diff --git a/src/sage/algebras/letterplace/free_algebra_element_letterplace.pyx b/src/sage/algebras/letterplace/free_algebra_element_letterplace.pyx
index b6bc0e3f87a..5c357f625ea 100644
--- a/src/sage/algebras/letterplace/free_algebra_element_letterplace.pyx
+++ b/src/sage/algebras/letterplace/free_algebra_element_letterplace.pyx
@@ -470,7 +470,7 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
 
     ################################
     ## Arithmetic
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         TEST::
 
@@ -484,7 +484,7 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
 
         """
         return FreeAlgebraElement_letterplace(self._parent,-self._poly,check=False)
-    cpdef ModuleElement _add_(self, ModuleElement other):
+    cpdef _add_(self, ModuleElement other):
         """
         Addition, under the side condition that either one summand
         is zero, or both summands have the same degree.
@@ -517,7 +517,7 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
         right._poly = A._current_ring(right._poly)
         return FreeAlgebraElement_letterplace(self._parent,self._poly+right._poly,check=False)
 
-    cpdef ModuleElement _sub_(self, ModuleElement other):
+    cpdef _sub_(self, ModuleElement other):
         """
         Difference, under the side condition that either one summand
         is zero or both have the same weighted degree.
@@ -556,7 +556,7 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
         right._poly = A._current_ring(right._poly)
         return FreeAlgebraElement_letterplace(self._parent,self._poly-right._poly,check=False)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Multiplication from the right with an element of the base ring.
 
@@ -570,7 +570,7 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
         """
         return FreeAlgebraElement_letterplace(self._parent,self._poly._lmul_(right),check=False)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Multiplication from the left with an element of the base ring.
 
@@ -584,7 +584,7 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
         """
         return FreeAlgebraElement_letterplace(self._parent,self._poly._rmul_(left),check=False)
 
-    cpdef RingElement _mul_(self, RingElement other):
+    cpdef _mul_(self, RingElement other):
         """
         Product of two free algebra elements in letterplace implementation.
 
diff --git a/src/sage/algebras/quatalg/quaternion_algebra_element.pyx b/src/sage/algebras/quatalg/quaternion_algebra_element.pyx
index 92752061629..0a3d7a30ed7 100644
--- a/src/sage/algebras/quatalg/quaternion_algebra_element.pyx
+++ b/src/sage/algebras/quatalg/quaternion_algebra_element.pyx
@@ -487,7 +487,7 @@ cdef class QuaternionAlgebraElement_abstract(AlgebraElement):
         """
         return ~self.reduced_norm() * self.conjugate()
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Return left*self, where left is in the base ring.
 
@@ -501,7 +501,7 @@ cdef class QuaternionAlgebraElement_abstract(AlgebraElement):
         """
         return self.__class__(self._parent, (left*self[0], left*self[1], left*self[2], left*self[3]), check=False)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Return self*right, where right is in the base ring.
 
@@ -515,7 +515,7 @@ cdef class QuaternionAlgebraElement_abstract(AlgebraElement):
         """
         return self.__class__(self._parent, (self[0]*right, self[1]*right, self[2]*right, self[3]*right), check=False)
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Return quotient of self by right.
 
@@ -724,7 +724,7 @@ cdef class QuaternionAlgebraElement_generic(QuaternionAlgebraElement_abstract):
         return (unpickle_QuaternionAlgebraElement_generic_v0,
                 (self._parent, (self.x, self.y, self.z, self.w)))
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Return the sum of self and _right.
 
@@ -740,7 +740,7 @@ cdef class QuaternionAlgebraElement_generic(QuaternionAlgebraElement_abstract):
         # TODO -- make this, etc. use __new__
         return QuaternionAlgebraElement_generic(self._parent, (self.x + right.x, self.y + right.y, self.z + right.z, self.w + right.w), check=False)
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Return the difference of self and _right.
 
@@ -755,7 +755,7 @@ cdef class QuaternionAlgebraElement_generic(QuaternionAlgebraElement_abstract):
         cdef QuaternionAlgebraElement_generic right = _right
         return QuaternionAlgebraElement_generic(self._parent, (self.x - right.x, self.y - right.y, self.z - right.z, self.w - right.w), check=False)
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         Return the product of self and _right.
 
@@ -1076,7 +1076,7 @@ cdef class QuaternionAlgebraElement_rational_field(QuaternionAlgebraElement_abst
         return (unpickle_QuaternionAlgebraElement_rational_field_v0,
                 (self._parent, (self[0], self[1], self[2], self[3])))
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         EXAMPLES::
 
@@ -1132,7 +1132,7 @@ cdef class QuaternionAlgebraElement_rational_field(QuaternionAlgebraElement_abst
         mpz_set(result.b, self.b)
         return result
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         EXAMPLES::
 
@@ -1173,7 +1173,7 @@ cdef class QuaternionAlgebraElement_rational_field(QuaternionAlgebraElement_abst
         mpz_set(result.b, self.b)
         return result
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         EXAMPLES::
 
@@ -1739,7 +1739,7 @@ cdef class QuaternionAlgebraElement_number_field(QuaternionAlgebraElement_abstra
         return (unpickle_QuaternionAlgebraElement_number_field_v0,
                 (self._parent, (self[0], self[1], self[2], self[3])))
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Add self and _right:
 
@@ -1810,7 +1810,7 @@ cdef class QuaternionAlgebraElement_number_field(QuaternionAlgebraElement_abstra
 
         return result
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Subtract _right from self.
 
@@ -1860,7 +1860,7 @@ cdef class QuaternionAlgebraElement_number_field(QuaternionAlgebraElement_abstra
 
         return result
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         Multiply self and _right.
 
diff --git a/src/sage/geometry/toric_lattice_element.pyx b/src/sage/geometry/toric_lattice_element.pyx
index 2a39ad678c2..984945b321a 100644
--- a/src/sage/geometry/toric_lattice_element.pyx
+++ b/src/sage/geometry/toric_lattice_element.pyx
@@ -316,7 +316,7 @@ cdef class ToricLatticeElement(Vector_integer_dense):
     # We need to override this function to prohibit default behaviour.
     # It seems to be called when right is in the same lattice as self, which
     # is wrong from our point of view.
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         """
         Raise a ``TypeError`` exception.
 
diff --git a/src/sage/groups/libgap_wrapper.pxd b/src/sage/groups/libgap_wrapper.pxd
index 22c2b4d0eb2..5ab5dad1776 100644
--- a/src/sage/groups/libgap_wrapper.pxd
+++ b/src/sage/groups/libgap_wrapper.pxd
@@ -5,8 +5,3 @@ from sage.libs.gap.element cimport GapElement
 cdef class ElementLibGAP(MultiplicativeGroupElement):
     cdef GapElement _libgap
     cpdef GapElement gap(self)
-    cpdef MonoidElement _mul_(left, MonoidElement right)
-    cpdef MultiplicativeGroupElement _div_(self, MultiplicativeGroupElement right)
-
-
-
diff --git a/src/sage/groups/libgap_wrapper.pyx b/src/sage/groups/libgap_wrapper.pyx
index 935d18dd6cd..53bae4075f9 100644
--- a/src/sage/groups/libgap_wrapper.pyx
+++ b/src/sage/groups/libgap_wrapper.pyx
@@ -540,7 +540,7 @@ cdef class ElementLibGAP(MultiplicativeGroupElement):
             from sage.misc.latex import latex
             return latex(self._repr_())
 
-    cpdef MonoidElement _mul_(left, MonoidElement right):
+    cpdef _mul_(left, MonoidElement right):
         """
         Multiplication of group elements
 
@@ -585,7 +585,7 @@ cdef class ElementLibGAP(MultiplicativeGroupElement):
         return cmp((<ElementLibGAP>left)._libgap,
                    (<ElementLibGAP>right)._libgap)
 
-    cpdef MultiplicativeGroupElement _div_(left, MultiplicativeGroupElement right):
+    cpdef _div_(left, MultiplicativeGroupElement right):
         """
         Division of group elements.
 
diff --git a/src/sage/groups/matrix_gps/group_element.pyx b/src/sage/groups/matrix_gps/group_element.pyx
index 0dd7a804dc9..db1064535e4 100644
--- a/src/sage/groups/matrix_gps/group_element.pyx
+++ b/src/sage/groups/matrix_gps/group_element.pyx
@@ -302,7 +302,7 @@ cdef class MatrixGroupElement_generic(MultiplicativeGroupElement):
         """
         return self._matrix
 
-    cpdef MonoidElement _mul_(self, MonoidElement other):
+    cpdef _mul_(self, MonoidElement other):
         """
         Return the product of ``self`` and`` other``, which must
         have identical parents.
diff --git a/src/sage/groups/perm_gps/permgroup_element.pyx b/src/sage/groups/perm_gps/permgroup_element.pyx
index 6165e340fbf..767731519d0 100644
--- a/src/sage/groups/perm_gps/permgroup_element.pyx
+++ b/src/sage/groups/perm_gps/permgroup_element.pyx
@@ -842,7 +842,7 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             elif is_Matrix(left):
                 return left.with_permuted_rows(self)
 
-    cpdef MonoidElement _mul_(left, MonoidElement _right):
+    cpdef _mul_(left, MonoidElement _right):
         """
         EXAMPLES::
 
diff --git a/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pyx b/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pyx
index 7f73bd0f00c..2d40f6f57e9 100644
--- a/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pyx
+++ b/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pyx
@@ -171,7 +171,7 @@ cdef class SemimonomialTransformation(MultiplicativeGroupElement):
         """
         return hash(self.v) + hash(self.perm) + hash(self.get_autom())
 
-    cpdef MonoidElement _mul_(left, MonoidElement _right):
+    cpdef _mul_(left, MonoidElement _right):
         r"""
         Multiplication of elements.
         
diff --git a/src/sage/libs/gap/element.pxd b/src/sage/libs/gap/element.pxd
index b28ead6e035..12c0c236b38 100644
--- a/src/sage/libs/gap/element.pxd
+++ b/src/sage/libs/gap/element.pxd
@@ -42,10 +42,6 @@ cdef class GapElement(RingElement):
 
     cdef _initialize(self, parent, libGAP_Obj obj)
     cpdef _type_number(self)
-    cpdef ModuleElement _add_(self, ModuleElement right)
-    cpdef ModuleElement _sub_(self, ModuleElement right)
-    cpdef RingElement _mul_(self, RingElement right)
-    cpdef RingElement _div_(self, RingElement right)
     cpdef is_bool(self)
 
 
diff --git a/src/sage/libs/gap/element.pyx b/src/sage/libs/gap/element.pyx
index 9efbe1b03f9..c00fa0e7958 100644
--- a/src/sage/libs/gap/element.pyx
+++ b/src/sage/libs/gap/element.pyx
@@ -690,7 +690,7 @@ cdef class GapElement(RingElement):
             libgap_exit()
         return result
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         r"""
         Add two GapElement objects.
 
@@ -723,7 +723,7 @@ cdef class GapElement(RingElement):
         return make_any_gap_element(self.parent(), result)
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         r"""
         Subtract two GapElement objects.
 
@@ -756,7 +756,7 @@ cdef class GapElement(RingElement):
         return make_any_gap_element(self.parent(), result)
 
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         r"""
         Multiply two GapElement objects.
 
@@ -789,7 +789,7 @@ cdef class GapElement(RingElement):
         return make_any_gap_element(self.parent(), result)
 
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         r"""
         Divide two GapElement objects.
 
diff --git a/src/sage/libs/pari/gen.pyx b/src/sage/libs/pari/gen.pyx
index 9514fb3cf72..2f2fbd849d0 100644
--- a/src/sage/libs/pari/gen.pyx
+++ b/src/sage/libs/pari/gen.pyx
@@ -240,19 +240,19 @@ cdef class gen(gen_auto):
         s = repr(self)
         return (objtogen, (s,))
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         sig_on()
         return P.new_gen(gadd(self.g, (<gen>right).g))
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         sig_on()
         return P.new_gen(gsub(self.g, (<gen> right).g))
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         sig_on()
         return P.new_gen(gmul(self.g, (<gen>right).g))
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         sig_on()
         return P.new_gen(gdiv(self.g, (<gen>right).g))
 
diff --git a/src/sage/matrix/matrix0.pyx b/src/sage/matrix/matrix0.pyx
index a444c00a22d..e99f6edc460 100644
--- a/src/sage/matrix/matrix0.pyx
+++ b/src/sage/matrix/matrix0.pyx
@@ -4592,7 +4592,7 @@ cdef class Matrix(sage.structure.element.Matrix):
     ###################################################
     # Arithmetic
     ###################################################
-    cdef Vector _vector_times_matrix_(self, Vector v):
+    cdef _vector_times_matrix_(self, Vector v):
         """
         Returns the vector times matrix product.
 
@@ -4650,7 +4650,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         return sum([v[i] * self.row(i, from_list=True)
                     for i in xrange(self._nrows)], M(0))
 
-    cdef Vector _matrix_times_vector_(self, Vector v):
+    cdef _matrix_times_vector_(self, Vector v):
         """
         EXAMPLES::
 
@@ -4760,7 +4760,7 @@ cdef class Matrix(sage.structure.element.Matrix):
             MS = self.matrix_space(n, m)
             return MS(X).transpose()
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Add two matrices with the same parent.
 
@@ -4783,7 +4783,7 @@ cdef class Matrix(sage.structure.element.Matrix):
                 A.set_unsafe(i,j,self.get_unsafe(i,j)._add_(right.get_unsafe(i,j)))
         return A
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Subtract two matrices with the same parent.
 
@@ -4849,7 +4849,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         return self.change_ring(self._base_ring.quotient_ring(p))
 
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -4886,7 +4886,7 @@ cdef class Matrix(sage.structure.element.Matrix):
                 ans.set_unsafe(r, c, x * self.get_unsafe(r, c))
         return ans
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES:
 
diff --git a/src/sage/matrix/matrix_cyclo_dense.pyx b/src/sage/matrix/matrix_cyclo_dense.pyx
index a8d5692adde..01bb3b6b9fa 100644
--- a/src/sage/matrix/matrix_cyclo_dense.pyx
+++ b/src/sage/matrix/matrix_cyclo_dense.pyx
@@ -469,7 +469,7 @@ cdef class Matrix_cyclo_dense(matrix_dense.Matrix_dense):
     #   * _dict -- sparse dictionary of underlying elements (need not be a copy)
     ########################################################################
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Return the sum of two dense cyclotomic matrices.
 
@@ -497,7 +497,7 @@ cdef class Matrix_cyclo_dense(matrix_dense.Matrix_dense):
         A._matrix = self._matrix + (<Matrix_cyclo_dense>right)._matrix
         return A
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Return the difference of two dense cyclotomic matrices.
 
@@ -524,7 +524,7 @@ cdef class Matrix_cyclo_dense(matrix_dense.Matrix_dense):
         A._matrix = self._matrix - (<Matrix_cyclo_dense>right)._matrix
         return A
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Multiply a dense cyclotomic matrix by a scalar.
 
@@ -566,7 +566,7 @@ cdef class Matrix_cyclo_dense(matrix_dense.Matrix_dense):
             A._matrix = T * self._matrix
         return A
 
-    cdef baseMatrix _matrix_times_matrix_(self, baseMatrix right):
+    cdef _matrix_times_matrix_(self, baseMatrix right):
         """
         Return the product of two cyclotomic dense matrices.
 
diff --git a/src/sage/matrix/matrix_double_dense.pyx b/src/sage/matrix/matrix_double_dense.pyx
index bc2e873fa08..b3977f5a7c2 100644
--- a/src/sage/matrix/matrix_double_dense.pyx
+++ b/src/sage/matrix/matrix_double_dense.pyx
@@ -317,7 +317,7 @@ cdef class Matrix_double_dense(matrix_dense.Matrix_dense):
     # LEVEL 2 functionality
     #   * def _pickle
     #   * def _unpickle
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two matrices together.
 
@@ -340,7 +340,7 @@ cdef class Matrix_double_dense(matrix_dense.Matrix_dense):
         M._matrix_numpy = _left._matrix_numpy + _right._matrix_numpy
         return M
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Return self - right
 
@@ -3581,7 +3581,7 @@ cdef class Matrix_double_dense(matrix_dense.Matrix_dense):
             posdef = self.fetch(cache_str)
         return posdef
 
-    cdef Vector _vector_times_matrix_(self,Vector v):
+    cdef _vector_times_matrix_(self,Vector v):
         if self._nrows == 0 or self._ncols == 0:
             return self._row_ambient_module().zero_vector()
         global numpy
@@ -3594,7 +3594,7 @@ cdef class Matrix_double_dense(matrix_dense.Matrix_dense):
         ans = numpy.dot(v_numpy,self._matrix_numpy)
         return M(ans)
 
-    cdef Vector _matrix_times_vector_(self,Vector v):
+    cdef _matrix_times_vector_(self,Vector v):
         if self._nrows == 0 or self._ncols == 0:
             return self._column_ambient_module().zero_vector()
 
diff --git a/src/sage/matrix/matrix_generic_sparse.pyx b/src/sage/matrix/matrix_generic_sparse.pyx
index 7994d4726a3..9199b3fdd9b 100644
--- a/src/sage/matrix/matrix_generic_sparse.pyx
+++ b/src/sage/matrix/matrix_generic_sparse.pyx
@@ -329,7 +329,7 @@ cdef class Matrix_generic_sparse(matrix_sparse.Matrix_sparse):
     # x  * _dict -- copy of the sparse dictionary of underlying elements
     ########################################################################
 
-    cpdef ModuleElement _add_(self, ModuleElement _other):
+    cpdef _add_(self, ModuleElement _other):
         """
         EXAMPLES::
 
diff --git a/src/sage/matrix/matrix_gf2e_dense.pyx b/src/sage/matrix/matrix_gf2e_dense.pyx
index 892ca5eb7dc..b9121fd3689 100644
--- a/src/sage/matrix/matrix_gf2e_dense.pyx
+++ b/src/sage/matrix/matrix_gf2e_dense.pyx
@@ -320,7 +320,7 @@ cdef class Matrix_gf2e_dense(matrix_dense.Matrix_dense):
         cdef int r = mzed_read_elem(self._entries, i, j)
         return word_to_poly(r, self._base_ring)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Return A+B
 
@@ -354,7 +354,7 @@ cdef class Matrix_gf2e_dense(matrix_dense.Matrix_dense):
 
         return A
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLE::
 
@@ -426,7 +426,7 @@ cdef class Matrix_gf2e_dense(matrix_dense.Matrix_dense):
         sig_off()
         return ans
 
-    cdef Matrix _matrix_times_matrix_(self, Matrix right):
+    cdef _matrix_times_matrix_(self, Matrix right):
         """
         Return A*B
 
@@ -633,7 +633,7 @@ cdef class Matrix_gf2e_dense(matrix_dense.Matrix_dense):
         sig_off()
         return ans
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Return ``a*B`` for ``a`` an element of the base field.
 
diff --git a/src/sage/matrix/matrix_gfpn_dense.pxd b/src/sage/matrix/matrix_gfpn_dense.pxd
index 34487536b14..a2374d6f32f 100644
--- a/src/sage/matrix/matrix_gfpn_dense.pxd
+++ b/src/sage/matrix/matrix_gfpn_dense.pxd
@@ -24,6 +24,6 @@ cdef class Matrix_gfpn_dense(Matrix_dense):
     cdef set_unsafe_int(self, Py_ssize_t i, Py_ssize_t j, int value)
     cdef inline int get_unsafe_int(self, Py_ssize_t i, Py_ssize_t j)
     cdef list _rowlist_(self, i, j=*)
-    cdef Matrix _matrix_times_matrix_(self, Matrix right)
+    cdef _matrix_times_matrix_(self, Matrix right)
     cpdef Matrix_gfpn_dense _multiply_classical(Matrix_gfpn_dense self, Matrix_gfpn_dense right)
     cpdef Matrix_gfpn_dense _multiply_strassen(Matrix_gfpn_dense self, Matrix_gfpn_dense right, cutoff=*)
diff --git a/src/sage/matrix/matrix_gfpn_dense.pyx b/src/sage/matrix/matrix_gfpn_dense.pyx
index 90d5198fb88..b9c53cf674f 100644
--- a/src/sage/matrix/matrix_gfpn_dense.pyx
+++ b/src/sage/matrix/matrix_gfpn_dense.pyx
@@ -1042,7 +1042,7 @@ cdef class Matrix_gfpn_dense(Matrix_dense):
         memcpy(MatGetPtr(OUT.Data, self.Data.Nor), other.Data.Data, FfCurrentRowSize*other.Data.Nor)
         return OUT
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         TESTS::
 
@@ -1066,7 +1066,7 @@ cdef class Matrix_gfpn_dense(Matrix_dense):
         MatAdd(Left.Data, Right.Data)
         return Left
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         TESTS::
 
@@ -1113,7 +1113,7 @@ cdef class Matrix_gfpn_dense(Matrix_dense):
             raise ValueError("The matrix must not be empty")
         return self._rmul_(self._base_ring(-1))
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -1136,7 +1136,7 @@ cdef class Matrix_gfpn_dense(Matrix_dense):
         MatMulScalar(OUT.Data, FfFromInt(self._converter.field_to_int(left)))
         return OUT
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -1232,8 +1232,10 @@ cdef class Matrix_gfpn_dense(Matrix_dense):
         sig_off()
         return OUT
 
-    cdef ModuleElement _mul_long(self, long n):
-        "multiply an MTX matrix with a field element represented by an integer"
+    cdef _mul_long(self, long n):
+        """
+        Multiply an MTX matrix with a field element represented by an integer
+        """
         if self.Data == NULL:
             raise ValueError("The matrix must not be empty")
         cdef Matrix_gfpn_dense left
diff --git a/src/sage/matrix/matrix_integer_dense.pyx b/src/sage/matrix/matrix_integer_dense.pyx
index 4b516620bf8..7adefd3d371 100644
--- a/src/sage/matrix/matrix_integer_dense.pyx
+++ b/src/sage/matrix/matrix_integer_dense.pyx
@@ -915,7 +915,7 @@ cdef class Matrix_integer_dense(matrix_dense.Matrix_dense):   # dense or sparse
         return M
 
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -935,7 +935,7 @@ cdef class Matrix_integer_dense(matrix_dense.Matrix_dense):   # dense or sparse
         sig_off()
         return M
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two dense matrices over ZZ.
 
@@ -960,7 +960,7 @@ cdef class Matrix_integer_dense(matrix_dense.Matrix_dense):   # dense or sparse
         sig_off()
         return M
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two dense matrices over ZZ.
 
@@ -1127,7 +1127,7 @@ cdef class Matrix_integer_dense(matrix_dense.Matrix_dense):   # dense or sparse
         return 0
 
     # TODO: Implement better
-    cdef Vector _vector_times_matrix_(self, Vector v):
+    cdef _vector_times_matrix_(self, Vector v):
         """
         Returns the vector times matrix product.
 
diff --git a/src/sage/matrix/matrix_integer_sparse.pyx b/src/sage/matrix/matrix_integer_sparse.pyx
index 7e1c4543d85..25521178738 100644
--- a/src/sage/matrix/matrix_integer_sparse.pyx
+++ b/src/sage/matrix/matrix_integer_sparse.pyx
@@ -175,7 +175,7 @@ cdef class Matrix_integer_sparse(matrix_sparse.Matrix_sparse):
     ########################################################################
     # def _pickle(self):
     # def _unpickle(self, data, int version):   # use version >= 0
-    # cpdef ModuleElement _add_(self, ModuleElement right):
+    # cpdef _add_(self, ModuleElement right):
     # cdef _mul_(self, Matrix right):
     # cpdef int _cmp_(self, Matrix right) except -2:
     # def __neg__(self):
@@ -184,7 +184,7 @@ cdef class Matrix_integer_sparse(matrix_sparse.Matrix_sparse):
     # def _multiply_classical(left, matrix.Matrix _right):
     # def _list(self):
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -206,7 +206,7 @@ cdef class Matrix_integer_sparse(matrix_sparse.Matrix_sparse):
             mpz_vector_scalar_multiply(M_row, self_row, _x.value)
         return M
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         cdef Py_ssize_t i, j
         cdef mpz_vector *self_row
         cdef mpz_vector *M_row
@@ -221,7 +221,7 @@ cdef class Matrix_integer_sparse(matrix_sparse.Matrix_sparse):
         mpz_clear(mul)
         return M
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         cdef Py_ssize_t i, j
         cdef mpz_vector *self_row
         cdef mpz_vector *M_row
diff --git a/src/sage/matrix/matrix_mod2_dense.pyx b/src/sage/matrix/matrix_mod2_dense.pyx
index 45c6e3fd660..3bef20ea33a 100644
--- a/src/sage/matrix/matrix_mod2_dense.pyx
+++ b/src/sage/matrix/matrix_mod2_dense.pyx
@@ -560,7 +560,7 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
     # def _pickle(self):
     # def _unpickle(self, data, int version):   # use version >= 0
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Matrix addition.
 
@@ -598,7 +598,7 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
 
         return A
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Matrix addition.
 
@@ -614,7 +614,7 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
         """
         return self._add_(right)
 
-    cdef Vector _matrix_times_vector_(self, Vector v):
+    cdef _matrix_times_vector_(self, Vector v):
         """
         EXAMPLES::
 
@@ -656,7 +656,7 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
             mzd_free(tmp)
         return c
 
-    cdef Matrix _matrix_times_matrix_(self, Matrix right):
+    cdef _matrix_times_matrix_(self, Matrix right):
         """
         Matrix multiplication.
 
diff --git a/src/sage/matrix/matrix_modn_dense_template.pxi b/src/sage/matrix/matrix_modn_dense_template.pxi
index ad24a8ef407..89d7d2f08b3 100644
--- a/src/sage/matrix/matrix_modn_dense_template.pxi
+++ b/src/sage/matrix/matrix_modn_dense_template.pxi
@@ -798,7 +798,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
         return M
 
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -808,7 +808,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
         """
         return self._rmul_(right)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -856,7 +856,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
         return A
 
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two dense matrices over `\Z/n\Z`
 
@@ -900,7 +900,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
         return M
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         r"""
         Subtract two dense matrices over `\Z/n\Z`
 
@@ -990,7 +990,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
         return 0
 
 
-    cdef Matrix _matrix_times_matrix_(self, Matrix right):
+    cdef _matrix_times_matrix_(self, Matrix right):
         """
         return ``self*right``
 
@@ -1191,7 +1191,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
 
         return ans
 
-    cdef Vector _vector_times_matrix_(self, Vector v):
+    cdef _vector_times_matrix_(self, Vector v):
         """
         ``v*self``
 
@@ -1248,7 +1248,7 @@ cdef class Matrix_modn_dense_template(matrix_dense.Matrix_dense):
         sig_free(_c)
         return c
 
-    cdef Vector _matrix_times_vector_(self, Vector v):
+    cdef _matrix_times_vector_(self, Vector v):
         """
         ``self*v``
 
diff --git a/src/sage/matrix/matrix_modn_sparse.pyx b/src/sage/matrix/matrix_modn_sparse.pyx
index cfac6f6f041..01af1b22bdb 100644
--- a/src/sage/matrix/matrix_modn_sparse.pyx
+++ b/src/sage/matrix/matrix_modn_sparse.pyx
@@ -255,7 +255,7 @@ cdef class Matrix_modn_sparse(matrix_sparse.Matrix_sparse):
     ########################################################################
     # def _pickle(self):
     # def _unpickle(self, data, int version):   # use version >= 0
-    # cpdef ModuleElement _add_(self, ModuleElement right):
+    # cpdef _add_(self, ModuleElement right):
     # cdef _mul_(self, Matrix right):
     # cpdef int _cmp_(self, Matrix right) except -2:
     # def __neg__(self):
diff --git a/src/sage/matrix/matrix_rational_dense.pyx b/src/sage/matrix/matrix_rational_dense.pyx
index 0344f09a56d..15f843f70f4 100644
--- a/src/sage/matrix/matrix_rational_dense.pyx
+++ b/src/sage/matrix/matrix_rational_dense.pyx
@@ -324,7 +324,7 @@ cdef class Matrix_rational_dense(matrix_dense.Matrix_dense):
     #   * _dict -- sparse dictionary of underlying elements (need not be a copy)
     ########################################################################
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -342,7 +342,7 @@ cdef class Matrix_rational_dense(matrix_dense.Matrix_dense):
             mpq_mul(M._entries[i], self._entries[i], _x.value)
         return M
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two dense matrices over QQ.
 
@@ -377,7 +377,7 @@ cdef class Matrix_rational_dense(matrix_dense.Matrix_dense):
         sig_off()
         return M
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two dense matrices over QQ.
 
@@ -427,7 +427,7 @@ cdef class Matrix_rational_dense(matrix_dense.Matrix_dense):
                         return 1
         return 0
 
-    cdef Vector _vector_times_matrix_(self, Vector v):
+    cdef _vector_times_matrix_(self, Vector v):
         """
         Returns the vector times matrix product.
 
diff --git a/src/sage/matrix/matrix_rational_sparse.pyx b/src/sage/matrix/matrix_rational_sparse.pyx
index 31241d078c1..b16075797d8 100644
--- a/src/sage/matrix/matrix_rational_sparse.pyx
+++ b/src/sage/matrix/matrix_rational_sparse.pyx
@@ -275,7 +275,7 @@ cdef class Matrix_rational_sparse(matrix_sparse.Matrix_sparse):
     ########################################################################
     # def _pickle(self):
     # def _unpickle(self, data, int version):   # use version >= 0
-    # cpdef ModuleElement _add_(self, ModuleElement right):
+    # cpdef _add_(self, ModuleElement right):
     # cdef _mul_(self, Matrix right):
     # cpdef int _cmp_(self, Matrix right) except -2:
     # def __neg__(self):
@@ -285,7 +285,7 @@ cdef class Matrix_rational_sparse(matrix_sparse.Matrix_sparse):
     # def _list(self):
 
 # TODO
-##     cpdef ModuleElement _lmul_(self, RingElement right):
+##     cpdef _lmul_(self, RingElement right):
 ##         """
 ##         EXAMPLES:
 ##             sage: a = matrix(QQ,2,range(6))
diff --git a/src/sage/matrix/matrix_sparse.pyx b/src/sage/matrix/matrix_sparse.pyx
index b16142e2d42..fc129ad1481 100644
--- a/src/sage/matrix/matrix_sparse.pyx
+++ b/src/sage/matrix/matrix_sparse.pyx
@@ -292,7 +292,7 @@ cdef class Matrix_sparse(matrix.Matrix):
 
         return left.new_matrix(left._nrows, right._ncols, entries=e, coerce=False, copy=False)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Left scalar multiplication. Internal usage only.
 
@@ -1007,7 +1007,7 @@ cdef class Matrix_sparse(matrix.Matrix):
             Z._subdivide_on_augment(self, other)
         return Z
 
-    cdef Vector _vector_times_matrix_(self, Vector v):
+    cdef _vector_times_matrix_(self, Vector v):
         """
         Returns the vector times matrix product.
 
@@ -1040,7 +1040,7 @@ cdef class Matrix_sparse(matrix.Matrix):
             s[j] += v[i] * a
         return s
 
-    cdef Vector _matrix_times_vector_(self, Vector v):
+    cdef _matrix_times_vector_(self, Vector v):
         """
         Returns the matrix times vector product.
 
diff --git a/src/sage/modular/arithgroup/arithgroup_element.pyx b/src/sage/modular/arithgroup/arithgroup_element.pyx
index 167c100dc63..d8c79e421d7 100644
--- a/src/sage/modular/arithgroup/arithgroup_element.pyx
+++ b/src/sage/modular/arithgroup/arithgroup_element.pyx
@@ -202,7 +202,7 @@ cdef class ArithmeticSubgroupElement(MultiplicativeGroupElement):
         """
         return True
 
-    cpdef MonoidElement _mul_(self, MonoidElement right):
+    cpdef _mul_(self, MonoidElement right):
         """
         Return self * right.
 
diff --git a/src/sage/modules/free_module_element.pyx b/src/sage/modules/free_module_element.pyx
index 8a2904801c2..bf827235275 100644
--- a/src/sage/modules/free_module_element.pyx
+++ b/src/sage/modules/free_module_element.pyx
@@ -2375,7 +2375,7 @@ cdef class FreeModuleElement(Vector):   # abstract base class
         else:
             return points(v, **kwds)
 
-    cpdef Element _dot_product_coerce_(left, Vector right):
+    cpdef _dot_product_coerce_(left, Vector right):
         """
         Return the dot product of left and right.
 
@@ -4097,7 +4097,7 @@ cdef class FreeModuleElement_generic_dense(FreeModuleElement):
 
     @cython.boundscheck(False)
     @cython.wraparound(False)
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         Add left and right.
 
@@ -4114,7 +4114,7 @@ cdef class FreeModuleElement_generic_dense(FreeModuleElement):
 
     @cython.boundscheck(False)
     @cython.wraparound(False)
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         """
         Subtract right from left.
 
@@ -4132,7 +4132,7 @@ cdef class FreeModuleElement_generic_dense(FreeModuleElement):
         v = [(<RingElement> a[i])._sub_(<RingElement> b[i]) for i in range(left._degree)]
         return left._new_c(v)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -4146,7 +4146,7 @@ cdef class FreeModuleElement_generic_dense(FreeModuleElement):
             v = [left * x for x in self._entries]
         return self._new_c(v)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -4164,7 +4164,7 @@ cdef class FreeModuleElement_generic_dense(FreeModuleElement):
 
     @cython.boundscheck(False)
     @cython.wraparound(False)
-    cpdef Vector _pairwise_product_(left, Vector right):
+    cpdef _pairwise_product_(left, Vector right):
         """
         EXAMPLES::
 
@@ -4540,7 +4540,7 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
                 entries = dict(entries)  # make a copy/convert to dict
         self._entries = entries
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         Add left and right.
 
@@ -4562,7 +4562,7 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
                 v[i] = a
         return left._new_c(v)
 
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         """
         EXAMPLES::
 
@@ -4582,7 +4582,7 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
                 v[i] = -a
         return left._new_c(v)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -4598,7 +4598,7 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
                     v[i] = prod
         return self._new_c(v)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -4614,7 +4614,7 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
                     v[i] = prod
         return self._new_c(v)
 
-    cpdef Element _dot_product_coerce_(left, Vector right):
+    cpdef _dot_product_coerce_(left, Vector right):
         """
         Return the dot product of left and right.
 
@@ -4666,7 +4666,7 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
                 z += a * e[i]
         return z
 
-    cpdef Vector _pairwise_product_(left, Vector right):
+    cpdef _pairwise_product_(left, Vector right):
         """
         EXAMPLES::
 
diff --git a/src/sage/modules/vector_double_dense.pyx b/src/sage/modules/vector_double_dense.pyx
index 0cae3fb2a89..760c327fadc 100644
--- a/src/sage/modules/vector_double_dense.pyx
+++ b/src/sage/modules/vector_double_dense.pyx
@@ -271,7 +271,7 @@ cdef class Vector_double_dense(FreeModuleElement):
                                                 numpy.PyArray_GETPTR1(self._vector_numpy, i)))
 
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two vectors together.
 
@@ -291,7 +291,7 @@ cdef class Vector_double_dense(FreeModuleElement):
 
         return self._new(_left._vector_numpy + _right._vector_numpy)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Return self - right
 
@@ -311,7 +311,7 @@ cdef class Vector_double_dense(FreeModuleElement):
 
         return self._new(_left._vector_numpy - _right._vector_numpy)
 
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         """
         Dot product of self and right.
 
@@ -335,7 +335,7 @@ cdef class Vector_double_dense(FreeModuleElement):
 
         return self._sage_dtype(numpy.dot(_left._vector_numpy, _right._vector_numpy))
 
-    cpdef Vector _pairwise_product_(self, Vector right):
+    cpdef _pairwise_product_(self, Vector right):
         """
         Return the component-wise product of self and right.
 
@@ -358,7 +358,7 @@ cdef class Vector_double_dense(FreeModuleElement):
 
         return self._new(_left._vector_numpy * _right._vector_numpy)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Multiply a scalar and vector
 
@@ -375,7 +375,7 @@ cdef class Vector_double_dense(FreeModuleElement):
         return self._new(self._python_dtype(left)*self._vector_numpy)
 
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Multiply a scalar and vector
 
diff --git a/src/sage/modules/vector_integer_dense.pyx b/src/sage/modules/vector_integer_dense.pyx
index 27d13bbb923..9860212e71d 100644
--- a/src/sage/modules/vector_integer_dense.pyx
+++ b/src/sage/modules/vector_integer_dense.pyx
@@ -213,7 +213,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
     def __reduce__(self):
         return (unpickle_v1, (self._parent, self.list(), self._degree, self._is_mutable))
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         cdef Vector_integer_dense z, r
         r = right
         z = self._new_c()
@@ -224,7 +224,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
         return z
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         cdef Vector_integer_dense z, r
         r = right
         z = self._new_c()
@@ -234,7 +234,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
             mpz_sub(z._entries[i], self._entries[i], r._entries[i])
         return z
 
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         """
         Dot product of dense vectors over the integers.
 
@@ -259,7 +259,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
         mpz_clear(t)
         return z
 
-    cpdef Vector _pairwise_product_(self, Vector right):
+    cpdef _pairwise_product_(self, Vector right):
         """
         EXAMPLES::
 
@@ -276,7 +276,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
             mpz_mul(z._entries[i], self._entries[i], r._entries[i])
         return z
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         cdef Vector_integer_dense z
         cdef Integer a
         a = left
@@ -287,7 +287,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
             mpz_mul(z._entries[i], self._entries[i], a.value)
         return z
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         cdef Vector_integer_dense z
         cdef Integer a
         a = right
@@ -298,7 +298,7 @@ cdef class Vector_integer_dense(free_module_element.FreeModuleElement):
             mpz_mul(z._entries[i], self._entries[i], a.value)
         return z
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         cdef Vector_integer_dense z
         z = self._new_c()
         cdef Py_ssize_t i
diff --git a/src/sage/modules/vector_mod2_dense.pyx b/src/sage/modules/vector_mod2_dense.pyx
index 6015ccc1821..1246b0f3248 100644
--- a/src/sage/modules/vector_mod2_dense.pyx
+++ b/src/sage/modules/vector_mod2_dense.pyx
@@ -263,7 +263,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
         """
         return unpickle_v0, (self._parent, self.list(), self._degree, self._is_mutable)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLE::
 
@@ -278,7 +278,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
             mzd_add(z._entries, self._entries, (<Vector_mod2_dense>right)._entries)
         return z
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLE::
 
@@ -309,7 +309,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
         return res
 
 
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         """
         EXAMPLES::
            sage: VS = VectorSpace(GF(2),3)
@@ -355,7 +355,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
 
         return n
 
-    cpdef Vector _pairwise_product_(self, Vector right):
+    cpdef _pairwise_product_(self, Vector right):
         """
         EXAMPLE::
 
@@ -375,7 +375,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
             z._entries.rows[0][i] = (self._entries.rows[0][i] & r._entries.rows[0][i])
         return z
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLE::
 
@@ -397,7 +397,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
             return self._new_c()
 
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLE::
 
@@ -413,7 +413,7 @@ cdef class Vector_mod2_dense(free_module_element.FreeModuleElement):
         """
         return self._rmul_(right)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         EXAMPLE::
 
diff --git a/src/sage/modules/vector_modn_dense.pyx b/src/sage/modules/vector_modn_dense.pyx
index a91d7afdd35..729daa1fd3f 100644
--- a/src/sage/modules/vector_modn_dense.pyx
+++ b/src/sage/modules/vector_modn_dense.pyx
@@ -256,7 +256,7 @@ cdef class Vector_modn_dense(free_module_element.FreeModuleElement):
     def __reduce__(self):
         return unpickle_v1, (self._parent, self.list(), self._degree, self._p, self._is_mutable)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         cdef Vector_modn_dense z, r
         r = right
         z = self._new_c()
@@ -266,7 +266,7 @@ cdef class Vector_modn_dense(free_module_element.FreeModuleElement):
         return z
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         cdef Vector_modn_dense z, r
         r = right
         z = self._new_c()
@@ -275,7 +275,7 @@ cdef class Vector_modn_dense(free_module_element.FreeModuleElement):
             z._entries[i] = (self._p + self._entries[i] - r._entries[i]) % self._p
         return z
 
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         cdef Py_ssize_t i
         cdef IntegerMod_int n
         cdef Vector_modn_dense r = right
@@ -288,7 +288,7 @@ cdef class Vector_modn_dense(free_module_element.FreeModuleElement):
 
         return n
 
-    cpdef Vector _pairwise_product_(self, Vector right):
+    cpdef _pairwise_product_(self, Vector right):
         """
         EXAMPLES:
            sage: v = vector(Integers(8), [2,3]); w = vector(Integers(8), [2,5])
@@ -305,7 +305,7 @@ cdef class Vector_modn_dense(free_module_element.FreeModuleElement):
             z._entries[i] = (self._entries[i] * r._entries[i]) % self._p
         return z
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         cdef Vector_modn_dense z
 
         cdef mod_int a = ivalue(left)
@@ -316,10 +316,10 @@ cdef class Vector_modn_dense(free_module_element.FreeModuleElement):
             z._entries[i] = (self._entries[i] * a) % self._p
         return z
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         return self._rmul_(right)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         cdef Vector_modn_dense z
         z = self._new_c()
         cdef Py_ssize_t i
diff --git a/src/sage/modules/vector_rational_dense.pyx b/src/sage/modules/vector_rational_dense.pyx
index 1433d9dbb8a..dcb00959b9a 100644
--- a/src/sage/modules/vector_rational_dense.pyx
+++ b/src/sage/modules/vector_rational_dense.pyx
@@ -242,7 +242,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
     def __reduce__(self):
         return (unpickle_v1, (self._parent, self.list(), self._degree, self._is_mutable))
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         cdef Vector_rational_dense z, r
         r = right
         z = self._new_c()
@@ -253,7 +253,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
         return z
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         cdef Vector_rational_dense z, r
         r = right
         z = self._new_c()
@@ -263,7 +263,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
             mpq_sub(z._entries[i], self._entries[i], r._entries[i])
         return z
 
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         """
         Dot product of dense vectors over the rationals.
 
@@ -289,7 +289,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
         return z
 
 
-    cpdef Vector _pairwise_product_(self, Vector right):
+    cpdef _pairwise_product_(self, Vector right):
         """
         EXAMPLES::
 
@@ -306,7 +306,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
             mpq_mul(z._entries[i], self._entries[i], r._entries[i])
         return z
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         cdef Vector_rational_dense z
         cdef Rational a
         if isinstance(left, Rational):
@@ -325,7 +325,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
         return z
 
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         cdef Vector_rational_dense z
         cdef Rational a
         if isinstance(right, Rational):
@@ -343,7 +343,7 @@ cdef class Vector_rational_dense(free_module_element.FreeModuleElement):
             mpq_mul(z._entries[i], self._entries[i], a.value)
         return z
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         cdef Vector_rational_dense z
         z = self._new_c()
         cdef Py_ssize_t i
diff --git a/src/sage/numerical/linear_functions.pxd b/src/sage/numerical/linear_functions.pxd
index c531b11a1fe..bf447586b6e 100644
--- a/src/sage/numerical/linear_functions.pxd
+++ b/src/sage/numerical/linear_functions.pxd
@@ -14,11 +14,6 @@ cdef class LinearFunctionsParent_class(Parent):
 cdef class LinearFunction(LinearFunctionOrConstraint):
     cdef dict _f
     cpdef iteritems(self)
-    cpdef ModuleElement _add_(self, ModuleElement b)
-    cpdef ModuleElement _sub_(self, ModuleElement b)
-    cpdef ModuleElement _lmul_(self, RingElement b)
-    cpdef ModuleElement _rmul_(self, RingElement b)
-    cpdef ModuleElement _neg_(self)
     cpdef _acted_upon_(self, x, bint self_on_left)
     cpdef is_zero(self)
     cpdef equals(LinearFunction left, LinearFunction right)
diff --git a/src/sage/numerical/linear_functions.pyx b/src/sage/numerical/linear_functions.pyx
index 0266a33030d..2d2498d917a 100644
--- a/src/sage/numerical/linear_functions.pyx
+++ b/src/sage/numerical/linear_functions.pyx
@@ -905,7 +905,7 @@ cdef class LinearFunction(LinearFunctionOrConstraint):
         except KeyError:
             return self.parent().base_ring().zero()
 
-    cpdef ModuleElement _add_(self, ModuleElement b):
+    cpdef _add_(self, ModuleElement b):
         r"""
         Defining the + operator
 
@@ -921,7 +921,7 @@ cdef class LinearFunction(LinearFunctionOrConstraint):
         P = self.parent()
         return P(e)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         r"""
         Defining the - operator (opposite).
 
@@ -934,7 +934,7 @@ cdef class LinearFunction(LinearFunctionOrConstraint):
         P = self.parent()
         return P(dict([(id,-coeff) for (id, coeff) in self._f.iteritems()]))
 
-    cpdef ModuleElement _sub_(self, ModuleElement b):
+    cpdef _sub_(self, ModuleElement b):
         r"""
         Defining the - operator (substraction).
 
@@ -952,7 +952,7 @@ cdef class LinearFunction(LinearFunctionOrConstraint):
         P = self.parent()
         return P(e)
 
-    cpdef ModuleElement _rmul_(self, RingElement b):
+    cpdef _rmul_(self, RingElement b):
         r"""
         Left multiplication by scalars
 
@@ -965,7 +965,7 @@ cdef class LinearFunction(LinearFunctionOrConstraint):
         P = self.parent()
         return P(dict([(id,b*coeff) for (id, coeff) in self._f.iteritems()]))
 
-    cpdef ModuleElement _lmul_(self, RingElement b):
+    cpdef _lmul_(self, RingElement b):
         r"""
         Right multiplication by scalars
 
diff --git a/src/sage/numerical/linear_tensor_element.pyx b/src/sage/numerical/linear_tensor_element.pyx
index 1fbd71116d5..5ee3c7bc70c 100644
--- a/src/sage/numerical/linear_tensor_element.pyx
+++ b/src/sage/numerical/linear_tensor_element.pyx
@@ -261,7 +261,7 @@ cdef class LinearTensor(ModuleElement):
             s += ']'
         return s
 
-    cpdef ModuleElement _add_(self, ModuleElement b):
+    cpdef _add_(self, ModuleElement b):
         r"""
         Return sum.
 
@@ -285,7 +285,7 @@ cdef class LinearTensor(ModuleElement):
             result[key] = self._f.get(key, 0) + coeff
         return self.parent()(result)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         r"""
         Return the negative.
 
@@ -305,7 +305,7 @@ cdef class LinearTensor(ModuleElement):
             result[key] = -coeff
         return self.parent()(result)
 
-    cpdef ModuleElement _sub_(self, ModuleElement b):
+    cpdef _sub_(self, ModuleElement b):
         r"""
         Return difference.
 
@@ -331,7 +331,7 @@ cdef class LinearTensor(ModuleElement):
             result[key] = self._f.get(key, 0) - coeff
         return self.parent()(result)
 
-    cpdef ModuleElement _rmul_(self, RingElement b):
+    cpdef _rmul_(self, RingElement b):
         r"""
         Return right multiplication by scalar.
 
diff --git a/src/sage/quivers/algebra_elements.pyx b/src/sage/quivers/algebra_elements.pyx
index e615fe4e8e0..9d9b03cd330 100644
--- a/src/sage/quivers/algebra_elements.pyx
+++ b/src/sage/quivers/algebra_elements.pyx
@@ -1014,7 +1014,7 @@ cdef class PathAlgebraElement(RingElement):
         return 1
 
     # negation
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         EXAMPLES::
 
@@ -1028,7 +1028,7 @@ cdef class PathAlgebraElement(RingElement):
         return self._new_(homog_poly_neg(self.data))
 
     # addition
-    cpdef ModuleElement _add_(self, ModuleElement other):
+    cpdef _add_(self, ModuleElement other):
         """
         EXAMPLES::
 
@@ -1105,7 +1105,7 @@ cdef class PathAlgebraElement(RingElement):
                     H1 = H1.nxt
                     H2 = H2.nxt
 
-    cpdef ModuleElement _sub_(self, ModuleElement other):
+    cpdef _sub_(self, ModuleElement other):
         """
         EXAMPLES::
 
@@ -1195,7 +1195,7 @@ cdef class PathAlgebraElement(RingElement):
 
 ## (scalar) multiplication
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -1226,7 +1226,7 @@ cdef class PathAlgebraElement(RingElement):
             return self._new_(outnxt)
         return self._new_(out)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -1298,7 +1298,7 @@ cdef class PathAlgebraElement(RingElement):
 
 ## Multiplication in the algebra
 
-    cpdef RingElement _mul_(self, RingElement  other):
+    cpdef _mul_(self, RingElement  other):
         """
         EXAMPLES::
 
diff --git a/src/sage/quivers/paths.pyx b/src/sage/quivers/paths.pyx
index 767386994a5..9d22ef35af2 100644
--- a/src/sage/quivers/paths.pyx
+++ b/src/sage/quivers/paths.pyx
@@ -427,7 +427,7 @@ cdef class QuiverPath(MonoidElement):
         for i in range(0,self._path.length):
             yield E[biseq_getitem(self._path, i)]
 
-    cpdef MonoidElement _mul_(self, MonoidElement other):
+    cpdef _mul_(self, MonoidElement other):
         """
         Compose two paths.
 
diff --git a/src/sage/rings/complex_arb.pyx b/src/sage/rings/complex_arb.pyx
index 895107646c4..5321077b55e 100644
--- a/src/sage/rings/complex_arb.pyx
+++ b/src/sage/rings/complex_arb.pyx
@@ -1748,7 +1748,7 @@ cdef class ComplexBall(RingElement):
         acb_conj(res.value, self.value)
         return res
 
-    cpdef ModuleElement _add_(self, ModuleElement other):
+    cpdef _add_(self, ModuleElement other):
         """
         Return the sum of two balls, rounded to the ambient field's precision.
 
@@ -1766,7 +1766,7 @@ cdef class ComplexBall(RingElement):
         if _do_sig(prec(self)): sig_off()
         return res
 
-    cpdef ModuleElement _sub_(self, ModuleElement other):
+    cpdef _sub_(self, ModuleElement other):
         """
         Return the difference of two balls, rounded to the ambient field's
         precision.
@@ -1807,7 +1807,7 @@ cdef class ComplexBall(RingElement):
         if _do_sig(prec(self)): sig_off()
         return res
 
-    cpdef RingElement _mul_(self, RingElement other):
+    cpdef _mul_(self, RingElement other):
         """
         Return the product of two balls, rounded to the ambient field's
         precision.
@@ -1907,7 +1907,7 @@ cdef class ComplexBall(RingElement):
             raise TypeError("unsupported operand type(s) for >>: '{}' and '{}'"
                             .format(type(val).__name__, type(shift).__name__))
 
-    cpdef RingElement _div_(self, RingElement other):
+    cpdef _div_(self, RingElement other):
         """
         Return the quotient of two balls, rounded to the ambient field's
         precision.
diff --git a/src/sage/rings/complex_double.pyx b/src/sage/rings/complex_double.pyx
index 4ea6cfd7102..35a2cec17fc 100644
--- a/src/sage/rings/complex_double.pyx
+++ b/src/sage/rings/complex_double.pyx
@@ -1155,7 +1155,7 @@ cdef class ComplexDoubleElement(FieldElement):
     # Arithmetic
     #######################################################################
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add ``self`` and ``right``.
 
@@ -1167,7 +1167,7 @@ cdef class ComplexDoubleElement(FieldElement):
         return self._new_c(gsl_complex_add(self._complex,
                                            (<ComplexDoubleElement>right)._complex))
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract ``self`` and ``right``.
 
@@ -1179,7 +1179,7 @@ cdef class ComplexDoubleElement(FieldElement):
         return self._new_c(gsl_complex_sub(self._complex,
                                 (<ComplexDoubleElement>right)._complex))
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply ``self`` and ``right``.
 
@@ -1191,7 +1191,7 @@ cdef class ComplexDoubleElement(FieldElement):
         return self._new_c(gsl_complex_mul(self._complex,
                        (<ComplexDoubleElement>right)._complex))
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide ``self`` by ``right``.
 
@@ -1225,7 +1225,7 @@ cdef class ComplexDoubleElement(FieldElement):
         """
         return self._new_c(gsl_complex_inverse(self._complex))
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         This function returns the negative of the complex number `z`:
 
diff --git a/src/sage/rings/complex_interval.pyx b/src/sage/rings/complex_interval.pyx
index a790abb6143..a01bda22d01 100644
--- a/src/sage/rings/complex_interval.pyx
+++ b/src/sage/rings/complex_interval.pyx
@@ -652,7 +652,7 @@ cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
         """
         return mpfi_has_zero(self.__re) and mpfi_has_zero(self.__im)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add ``self`` and ``right``.
 
@@ -667,7 +667,7 @@ cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
         mpfi_add(x.__im, self.__im, (<ComplexIntervalFieldElement>right).__im)
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract ``self`` by ``right``.
 
@@ -682,7 +682,7 @@ cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
         mpfi_sub(x.__im, self.__im, (<ComplexIntervalFieldElement>right).__im)
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply ``self`` and ``right``.
 
@@ -777,7 +777,7 @@ cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
         mpfi_clear(t1)
         return x
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide ``self`` by ``right``.
 
diff --git a/src/sage/rings/complex_mpc.pyx b/src/sage/rings/complex_mpc.pyx
index 0256a3946fe..b2df460d17d 100644
--- a/src/sage/rings/complex_mpc.pyx
+++ b/src/sage/rings/complex_mpc.pyx
@@ -1293,7 +1293,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
     # Basic Arithmetic
     ################################
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two complex numbers with the same parent.
 
@@ -1308,7 +1308,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         mpc_add(z.value, self.value, (<MPComplexNumber>right).value, (<MPComplexField_class>self._parent).__rnd)
         return z
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two complex numbers with the same parent.
 
@@ -1323,7 +1323,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         mpc_sub(z.value, self.value, (<MPComplexNumber>right).value, (<MPComplexField_class>self._parent).__rnd)
         return z
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply two complex numbers with the same parent.
 
@@ -1338,7 +1338,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         mpc_mul(z.value, self.value, (<MPComplexNumber>right).value, (<MPComplexField_class>self._parent).__rnd)
         return z
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide two complex numbers with the same parent.
 
@@ -1357,7 +1357,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
             mpc_div(z.value, self.value, x.value, (<MPComplexField_class>self._parent).__rnd)
         return z
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Return the negative of this complex number.
 
diff --git a/src/sage/rings/complex_number.pyx b/src/sage/rings/complex_number.pyx
index d5c70b47206..c1a7dc908b8 100644
--- a/src/sage/rings/complex_number.pyx
+++ b/src/sage/rings/complex_number.pyx
@@ -596,7 +596,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         im = mpfr_to_mpfval(self.__im)
         return make_mpc((re, im))
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add ``self`` to ``right``.
 
@@ -611,7 +611,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_add(x.__im, self.__im, (<ComplexNumber>right).__im, rnd)
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract ``right`` from ``self``.
 
@@ -626,7 +626,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_sub(x.__im, self.__im, (<ComplexNumber>right).__im, rnd)
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply ``self`` by ``right``.
 
@@ -730,7 +730,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(t1)
         return x
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide ``self`` by ``right``.
 
diff --git a/src/sage/rings/finite_rings/element_givaro.pyx b/src/sage/rings/finite_rings/element_givaro.pyx
index c20c88f506f..495fd7f22df 100644
--- a/src/sage/rings/finite_rings/element_givaro.pyx
+++ b/src/sage/rings/finite_rings/element_givaro.pyx
@@ -1088,7 +1088,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         else:
             raise ValueError("must be a perfect square.")
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two elements.
 
@@ -1103,7 +1103,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
                                               (<FiniteField_givaroElement>right).element )
         return make_FiniteField_givaroElement(self._cache,r)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply two elements.
 
@@ -1121,7 +1121,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         return make_FiniteField_givaroElement(self._cache,r)
 
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide two elements
 
@@ -1143,7 +1143,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
                                               (<FiniteField_givaroElement>right).element)
         return make_FiniteField_givaroElement(self._cache,r)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two elements.
 
diff --git a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
index 40e3e1f287a..c129bb2c523 100644
--- a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
+++ b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
@@ -675,7 +675,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         else:
             return a
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two elements.
 
@@ -691,7 +691,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         GF2E_add(r.x, (<FiniteField_ntl_gf2eElement>self).x, (<FiniteField_ntl_gf2eElement>right).x)
         return r
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply two elements.
 
@@ -707,7 +707,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         GF2E_mul(r.x, (<FiniteField_ntl_gf2eElement>self).x, (<FiniteField_ntl_gf2eElement>right).x)
         return r
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide two elements.
 
@@ -729,7 +729,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         GF2E_div(r.x, (<FiniteField_ntl_gf2eElement>self).x, (<FiniteField_ntl_gf2eElement>right).x)
         return r
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two elements.
 
diff --git a/src/sage/rings/finite_rings/element_pari_ffelt.pyx b/src/sage/rings/finite_rings/element_pari_ffelt.pyx
index 004b98eb1ea..a0487bc3902 100644
--- a/src/sage/rings/finite_rings/element_pari_ffelt.pyx
+++ b/src/sage/rings/finite_rings/element_pari_ffelt.pyx
@@ -443,7 +443,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
         sig_off()
         return r
 
-    cpdef ModuleElement _add_(FiniteFieldElement_pari_ffelt self, ModuleElement right):
+    cpdef _add_(FiniteFieldElement_pari_ffelt self, ModuleElement right):
         """
         Addition.
 
@@ -459,7 +459,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
                            (<FiniteFieldElement_pari_ffelt>right).val))
         return x
 
-    cpdef ModuleElement _sub_(FiniteFieldElement_pari_ffelt self, ModuleElement right):
+    cpdef _sub_(FiniteFieldElement_pari_ffelt self, ModuleElement right):
         """
         Subtraction.
 
@@ -475,7 +475,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
                            (<FiniteFieldElement_pari_ffelt>right).val))
         return x
 
-    cpdef RingElement _mul_(FiniteFieldElement_pari_ffelt self, RingElement right):
+    cpdef _mul_(FiniteFieldElement_pari_ffelt self, RingElement right):
         """
         Multiplication.
 
@@ -491,7 +491,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
                            (<FiniteFieldElement_pari_ffelt>right).val))
         return x
 
-    cpdef RingElement _div_(FiniteFieldElement_pari_ffelt self, RingElement right):
+    cpdef _div_(FiniteFieldElement_pari_ffelt self, RingElement right):
         """
         Division.
 
diff --git a/src/sage/rings/finite_rings/integer_mod.pyx b/src/sage/rings/finite_rings/integer_mod.pyx
index df7fac47058..3c91d80e93e 100644
--- a/src/sage/rings/finite_rings/integer_mod.pyx
+++ b/src/sage/rings/finite_rings/integer_mod.pyx
@@ -1651,7 +1651,7 @@ cdef class IntegerMod_abstract(FiniteRingElement):
                 return infinity
         return r
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Exact division for prime moduli, for compatibility with other fields.
 
@@ -1909,7 +1909,7 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
         mpz_set(x.value, self.value)
         return x
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -1924,7 +1924,7 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
             mpz_sub(x.value, x.value, self.__modulus.sageInteger.value)
         return x;
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -1939,7 +1939,7 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
             mpz_add(x.value, x.value, self.__modulus.sageInteger.value)
         return x;
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         EXAMPLES::
 
@@ -1955,7 +1955,7 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
         mpz_sub(x.value, self.__modulus.sageInteger.value, self.value)
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -1969,7 +1969,7 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
         mpz_fdiv_r(x.value, x.value, self.__modulus.sageInteger.value)
         return x
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -2324,7 +2324,7 @@ cdef class IntegerMod_int(IntegerMod_abstract):
         x.ivalue = self.ivalue
         return x
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -2338,7 +2338,7 @@ cdef class IntegerMod_int(IntegerMod_abstract):
             x = x - self.__modulus.int32
         return self._new_c(x)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -2352,7 +2352,7 @@ cdef class IntegerMod_int(IntegerMod_abstract):
             x = x + self.__modulus.int32
         return self._new_c(x)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         EXAMPLES::
 
@@ -2365,7 +2365,7 @@ cdef class IntegerMod_int(IntegerMod_abstract):
             return self
         return self._new_c(self.__modulus.int32 - self.ivalue)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -2375,7 +2375,7 @@ cdef class IntegerMod_int(IntegerMod_abstract):
         """
         return self._new_c((self.ivalue * (<IntegerMod_int>right).ivalue) % self.__modulus.int32)
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -3152,7 +3152,7 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
     def __copy__(IntegerMod_int64 self):
         return self._new_c(self.ivalue)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -3166,7 +3166,7 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
             x = x - self.__modulus.int64
         return self._new_c(x)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -3180,7 +3180,7 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
             x = x + self.__modulus.int64
         return self._new_c(x)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         EXAMPLES::
 
@@ -3193,7 +3193,7 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
             return self
         return self._new_c(self.__modulus.int64 - self.ivalue)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -3204,7 +3204,7 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
         return self._new_c((self.ivalue * (<IntegerMod_int64>right).ivalue) % self.__modulus.int64)
 
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         EXAMPLES::
 
diff --git a/src/sage/rings/fraction_field_FpT.pyx b/src/sage/rings/fraction_field_FpT.pyx
index 3ae3535e31e..8b0959ecd6a 100644
--- a/src/sage/rings/fraction_field_FpT.pyx
+++ b/src/sage/rings/fraction_field_FpT.pyx
@@ -443,7 +443,7 @@ cdef class FpTElement(RingElement):
         nmod_poly_swap(x._numer, x._denom)
         return x
 
-    cpdef ModuleElement _add_(self, ModuleElement _other):
+    cpdef _add_(self, ModuleElement _other):
         """
         Returns the sum of this fraction field element and another.
 
@@ -471,7 +471,7 @@ cdef class FpTElement(RingElement):
         normalize(x._numer, x._denom, self.p)
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement _other):
+    cpdef _sub_(self, ModuleElement _other):
         """
         Returns the difference of this fraction field element and another.
 
@@ -493,7 +493,7 @@ cdef class FpTElement(RingElement):
         normalize(x._numer, x._denom, self.p)
         return x
 
-    cpdef RingElement _mul_(self, RingElement _other):
+    cpdef _mul_(self, RingElement _other):
         """
         Returns the product of this fraction field element and another.
 
@@ -513,7 +513,7 @@ cdef class FpTElement(RingElement):
         normalize(x._numer, x._denom, self.p)
         return x
 
-    cpdef RingElement _div_(self, RingElement _other):
+    cpdef _div_(self, RingElement _other):
         """
         Returns the quotient of this fraction field element and another.
 
diff --git a/src/sage/rings/fraction_field_element.pyx b/src/sage/rings/fraction_field_element.pyx
index e3ae4979bc6..8dd9d67a166 100644
--- a/src/sage/rings/fraction_field_element.pyx
+++ b/src/sage/rings/fraction_field_element.pyx
@@ -484,7 +484,7 @@ cdef class FractionFieldElement(FieldElement):
 
         return s
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Computes the sum of ``self`` and ``right``.
 
@@ -563,7 +563,7 @@ cdef class FractionFieldElement(FieldElement):
         return self.__class__(self._parent, rnum*sden + rden*snum, rden*sden,
             coerce=False, reduce=False)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Computes the difference of ``self`` and ``right``.
 
@@ -583,7 +583,7 @@ cdef class FractionFieldElement(FieldElement):
         """
         return self._add_(-right)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Computes the product of ``self`` and ``right``.
 
@@ -648,7 +648,7 @@ cdef class FractionFieldElement(FieldElement):
         return self.__class__(self._parent, rnum * snum, rden * sden,
             coerce=False, reduce=False)
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Computes the quotient of ``self`` and ``right``.
 
diff --git a/src/sage/rings/function_field/function_field_element.pyx b/src/sage/rings/function_field/function_field_element.pyx
index 027407f07c3..9ee0d77cd7e 100644
--- a/src/sage/rings/function_field/function_field_element.pyx
+++ b/src/sage/rings/function_field/function_field_element.pyx
@@ -387,7 +387,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         cdef FunctionFieldElement right = <FunctionFieldElement>other
         return cmp(left._x, right._x)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -404,7 +404,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         res._x = self._x + (<FunctionFieldElement>right)._x
         return res
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -419,7 +419,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         res._x = self._x - (<FunctionFieldElement>right)._x
         return res
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -432,7 +432,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         res._x = (self._x * (<FunctionFieldElement>right)._x) % self._parent.polynomial()
         return res
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -604,7 +604,7 @@ cdef class FunctionFieldElement_rational(FunctionFieldElement):
         c = cmp(left._parent, right._parent)
         return c or cmp(left._x, right._x)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -616,7 +616,7 @@ cdef class FunctionFieldElement_rational(FunctionFieldElement):
         res._x = self._x + (<FunctionFieldElement>right)._x
         return res
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLES::
 
@@ -628,7 +628,7 @@ cdef class FunctionFieldElement_rational(FunctionFieldElement):
         res._x = self._x - (<FunctionFieldElement>right)._x
         return res
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -640,7 +640,7 @@ cdef class FunctionFieldElement_rational(FunctionFieldElement):
         res._x = self._x * (<FunctionFieldElement>right)._x
         return res
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         EXAMPLES::
 
diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index e40042f7061..d8534ea3948 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -1552,7 +1552,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         mpz_to_ZZ(z, self.value)
         sig_off()
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Integer addition.
 
@@ -1572,7 +1572,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         mpz_add(x.value, self.value, (<Integer>right).value)
         return x
 
-    cdef RingElement _add_long(self, long n):
+    cdef _add_long(self, long n):
         """
         Fast path for adding a C long.
 
@@ -1611,7 +1611,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             mpz_sub_ui(x.value, self.value, 0 - <unsigned long>n)
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Integer subtraction.
 
@@ -1649,7 +1649,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         mpz_neg(x.value, self.value)
         return x
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         cdef Integer x = <Integer>PY_NEW(Integer)
         mpz_neg(x.value, self.value)
         return x
@@ -1669,7 +1669,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             else:
                 return s * int(self) # will raise the appropriate exception
 
-    cdef ModuleElement _mul_long(self, long n):
+    cdef _mul_long(self, long n):
         """
         Fast path for multiplying a C long.
 
@@ -1691,7 +1691,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             mpz_mul_si(x.value, self.value, n)
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Integer multiplication.
 
@@ -1716,7 +1716,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             mpz_mul(x.value, self.value, (<Integer>right).value)
         return x
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         r"""
         Computes `\frac{a}{b}`
 
@@ -1732,7 +1732,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         # we can't cimport rationals.
         return the_integer_ring._div(self, right)
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         r"""
         Computes the whole part of `\frac{x}{y}`.
 
diff --git a/src/sage/rings/laurent_series_ring_element.pyx b/src/sage/rings/laurent_series_ring_element.pyx
index 124794a490f..db46cf9a84a 100644
--- a/src/sage/rings/laurent_series_ring_element.pyx
+++ b/src/sage/rings/laurent_series_ring_element.pyx
@@ -549,7 +549,7 @@ cdef class LaurentSeries(AlgebraElement):
                 self.__u = self.__u._parent(coeffs)
         self.__normalize()
 
-    cpdef ModuleElement _add_(self, ModuleElement right_m):
+    cpdef _add_(self, ModuleElement right_m):
         """
         Add two power series with the same parent.
 
@@ -599,7 +599,7 @@ cdef class LaurentSeries(AlgebraElement):
         # 3. Add
         return LaurentSeries(self._parent, f1 + f2, m)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right_m):
+    cpdef _sub_(self, ModuleElement right_m):
         """
         Subtract two power series with the same parent.
 
@@ -684,7 +684,7 @@ cdef class LaurentSeries(AlgebraElement):
         """
         return LaurentSeries(self._parent, -self.__u, self.__n)
 
-    cpdef RingElement _mul_(self, RingElement right_r):
+    cpdef _mul_(self, RingElement right_r):
         """
         EXAMPLES::
 
@@ -699,10 +699,10 @@ cdef class LaurentSeries(AlgebraElement):
                              self.__u * right.__u,
                              self.__n + right.__n)
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         return LaurentSeries(self._parent, self.__u._rmul_(c), self.__n)
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         return LaurentSeries(self._parent, self.__u._lmul_(c), self.__n)
 
     def __pow__(_self, r, dummy):
@@ -809,7 +809,7 @@ cdef class LaurentSeries(AlgebraElement):
         """
         return LaurentSeries(self._parent, self.__u >> (n - self.__n), n)
 
-    cpdef RingElement _div_(self, RingElement right_r):
+    cpdef _div_(self, RingElement right_r):
         """
         EXAMPLES::
 
diff --git a/src/sage/rings/number_field/number_field_element.pxd b/src/sage/rings/number_field/number_field_element.pxd
index 8b27454cc51..637ccb8b214 100644
--- a/src/sage/rings/number_field/number_field_element.pxd
+++ b/src/sage/rings/number_field/number_field_element.pxd
@@ -28,9 +28,6 @@ cdef class NumberFieldElement(FieldElement):
 
     cdef void _invert_c_(self, ZZX_c *num, ZZ_c *den)
     cdef void _reduce_c_(self)
-    cpdef ModuleElement _add_(self, ModuleElement right)
-    cpdef ModuleElement _sub_(self, ModuleElement right)
-    cpdef ModuleElement _neg_(self)
 
     cpdef list _coefficients(self)
 
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index 15bb506f108..0c2a57a4afc 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -2003,7 +2003,7 @@ cdef class NumberFieldElement(FieldElement):
         self.__numerator = t2
         self.__denominator = t1
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         r"""
         EXAMPLE::
 
@@ -2024,7 +2024,7 @@ cdef class NumberFieldElement(FieldElement):
         x._reduce_c_()
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         r"""
         EXAMPLES::
 
@@ -2043,7 +2043,7 @@ cdef class NumberFieldElement(FieldElement):
         x._reduce_c_()
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Returns the product of self and other as elements of a number
         field.
@@ -2092,7 +2092,7 @@ cdef class NumberFieldElement(FieldElement):
         # but asymptotically fast poly multiplication means it's
         # actually faster to *not* build a table!?!
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Returns the quotient of self and other as elements of a number
         field.
@@ -2168,7 +2168,7 @@ cdef class NumberFieldElement(FieldElement):
         """
         return not IsZero_ZZX(self.__numerator)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         r"""
         EXAMPLE::
 
@@ -4556,7 +4556,7 @@ cdef class OrderElement_absolute(NumberFieldElement_absolute):
         """
         return self._number_field
 
-    cpdef RingElement _div_(self, RingElement other):
+    cpdef _div_(self, RingElement other):
         r"""
         Implement division, checking that the result has the right parent.
 
@@ -4684,7 +4684,7 @@ cdef class OrderElement_relative(NumberFieldElement_relative):
         x.__fld_denominator = self.__fld_denominator
         return x
 
-    cpdef RingElement _div_(self, RingElement other):
+    cpdef _div_(self, RingElement other):
         r"""
         Implement division, checking that the result has the right parent.
 
diff --git a/src/sage/rings/number_field/number_field_element_quadratic.pyx b/src/sage/rings/number_field/number_field_element_quadratic.pyx
index 9793a3407f7..62645c1ed3f 100644
--- a/src/sage/rings/number_field/number_field_element_quadratic.pyx
+++ b/src/sage/rings/number_field/number_field_element_quadratic.pyx
@@ -960,7 +960,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         mpz_clear(gcd)
 
 
-    cpdef ModuleElement _add_(self, ModuleElement other_m):
+    cpdef _add_(self, ModuleElement other_m):
         """
         EXAMPLES::
 
@@ -1017,7 +1017,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         return res
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement other_m):
+    cpdef _sub_(self, ModuleElement other_m):
         """
         EXAMPLES::
 
@@ -1084,7 +1084,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         return res
 
 
-    cpdef RingElement _mul_(self, RingElement other_m):
+    cpdef _mul_(self, RingElement other_m):
         """
         EXAMPLES:
             sage: K.<a> = NumberField(x^2+23)
@@ -1141,7 +1141,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         return res
 
 
-    cpdef ModuleElement _rmul_(self, RingElement _c):
+    cpdef _rmul_(self, RingElement _c):
         """
         EXAMPLE:
             sage: K.<a> = NumberField(x^2+43)
@@ -1157,7 +1157,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         return res
 
 
-    cpdef ModuleElement _lmul_(self, RingElement _c):
+    cpdef _lmul_(self, RingElement _c):
         """
         EXAMPLE:
             sage: K.<a> = NumberField(x^2+43)
@@ -1173,7 +1173,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         return res
 
 
-    cpdef RingElement _div_(self, RingElement other):
+    cpdef _div_(self, RingElement other):
         """
         EXAMPLES:
             sage: K.<a> = NumberField(x^2-5)
@@ -2084,7 +2084,7 @@ cdef class OrderElement_quadratic(NumberFieldElement_quadratic):
         return self._parent.number_field()
 
     # We must override these since the basering is now ZZ not QQ.
-    cpdef ModuleElement _rmul_(self, RingElement _c):
+    cpdef _rmul_(self, RingElement _c):
         """
         EXAMPLE:
             sage: K.<a> = NumberField(x^2-27)
@@ -2103,7 +2103,7 @@ cdef class OrderElement_quadratic(NumberFieldElement_quadratic):
         return res
 
 
-    cpdef ModuleElement _lmul_(self, RingElement _c):
+    cpdef _lmul_(self, RingElement _c):
         """
         EXAMPLE:
             sage: K.<a> = NumberField(x^2+43)
@@ -2121,7 +2121,7 @@ cdef class OrderElement_quadratic(NumberFieldElement_quadratic):
         res._reduce_c_()
         return res
 
-    cpdef RingElement _div_(self, RingElement other):
+    cpdef _div_(self, RingElement other):
         r"""
         Implement division, checking that the result has the
         right parent. It's not so crucial what the parent actually
diff --git a/src/sage/rings/padics/CA_template.pxi b/src/sage/rings/padics/CA_template.pxi
index 26557694de0..a177e9418e2 100644
--- a/src/sage/rings/padics/CA_template.pxi
+++ b/src/sage/rings/padics/CA_template.pxi
@@ -163,7 +163,7 @@ cdef class CAElement(pAdicTemplateElement):
         """
         return unpickle_cae_v2, (self.__class__, self.parent(), cpickle(self.value, self.prime_pow), self.absprec)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Return the additive inverse of this element.
 
@@ -180,7 +180,7 @@ cdef class CAElement(pAdicTemplateElement):
         creduce_small(ans.value, ans.value, ans.absprec, ans.prime_pow)
         return ans
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Return the sum of this element and ``_right``.
 
@@ -204,7 +204,7 @@ cdef class CAElement(pAdicTemplateElement):
         creduce(ans.value, ans.value, ans.absprec, ans.prime_pow)
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Return the difference of this element and ``_right``.
 
@@ -246,7 +246,7 @@ cdef class CAElement(pAdicTemplateElement):
         """
         return ~self.parent().fraction_field()(self)
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         Return the product of this element and ``_right``.
 
@@ -269,7 +269,7 @@ cdef class CAElement(pAdicTemplateElement):
         creduce(ans.value, ans.value, ans.absprec, ans.prime_pow)
         return ans
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Return the quotient of this element and ``right``.
 
diff --git a/src/sage/rings/padics/CR_template.pxi b/src/sage/rings/padics/CR_template.pxi
index a919df34d86..205ea23d999 100644
--- a/src/sage/rings/padics/CR_template.pxi
+++ b/src/sage/rings/padics/CR_template.pxi
@@ -273,7 +273,7 @@ cdef class CRElement(pAdicTemplateElement):
         """
         return unpickle_cre_v2, (self.__class__, self.parent(), cpickle(self.unit, self.prime_pow), self.ordp, self.relprec)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Return the additive inverse of this element.
 
@@ -297,7 +297,7 @@ cdef class CRElement(pAdicTemplateElement):
             creduce(ans.unit, ans.unit, ans.relprec, ans.prime_pow)
         return ans
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Return the sum of this element and ``_right``.
 
@@ -339,7 +339,7 @@ cdef class CRElement(pAdicTemplateElement):
                 creduce(ans.unit, ans.unit, ans.relprec, ans.prime_pow)
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Return the difference of this element and ``_right``.
 
@@ -413,7 +413,7 @@ cdef class CRElement(pAdicTemplateElement):
         cinvert(ans.unit, self.unit, ans.relprec, ans.prime_pow)
         return ans
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         r"""
         Return the product of this element and ``_right``.
 
@@ -444,7 +444,7 @@ cdef class CRElement(pAdicTemplateElement):
         check_ordp(ans.ordp)
         return ans
 
-    cpdef RingElement _div_(self, RingElement _right):
+    cpdef _div_(self, RingElement _right):
         """
         Return the quotient of this element and ``right``.
 
@@ -758,7 +758,7 @@ cdef class CRElement(pAdicTemplateElement):
                 ans._normalize()
         return ans
 
-    cpdef RingElement _floordiv_(self, RingElement _right):
+    cpdef _floordiv_(self, RingElement _right):
         """
         Floor division.
 
diff --git a/src/sage/rings/padics/FM_template.pxi b/src/sage/rings/padics/FM_template.pxi
index ac1e1bc3b8d..d033b1965e2 100644
--- a/src/sage/rings/padics/FM_template.pxi
+++ b/src/sage/rings/padics/FM_template.pxi
@@ -158,7 +158,7 @@ cdef class FMElement(pAdicTemplateElement):
         """
         return unpickle_fme_v2, (self.__class__, self.parent(), cpickle(self.value, self.prime_pow))
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         r"""
         Return the additive inverse of this element.
 
@@ -173,7 +173,7 @@ cdef class FMElement(pAdicTemplateElement):
         creduce_small(ans.value, ans.value, ans.prime_pow.prec_cap, ans.prime_pow)
         return ans
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         r"""
         Return the sum of this element and ``_right``.
 
@@ -193,7 +193,7 @@ cdef class FMElement(pAdicTemplateElement):
         creduce_small(ans.value, ans.value, ans.prime_pow.prec_cap, ans.prime_pow)
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         r"""
         Return the difference of this element and ``_right``.
 
@@ -238,7 +238,7 @@ cdef class FMElement(pAdicTemplateElement):
         cinvert(ans.value, self.value, ans.prime_pow.prec_cap, ans.prime_pow)
         return ans
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         r"""
         Return the product of this element and ``_right``.
 
@@ -256,7 +256,7 @@ cdef class FMElement(pAdicTemplateElement):
         creduce(ans.value, ans.value, ans.prime_pow.prec_cap, ans.prime_pow)
         return ans
 
-    cpdef RingElement _div_(self, RingElement _right):
+    cpdef _div_(self, RingElement _right):
         r"""
         Return the quotient of this element and ``right``. ``right`` must have
         valuation zero.
diff --git a/src/sage/rings/padics/local_generic_element.pxd b/src/sage/rings/padics/local_generic_element.pxd
index dc563f267bc..9111d2f173c 100644
--- a/src/sage/rings/padics/local_generic_element.pxd
+++ b/src/sage/rings/padics/local_generic_element.pxd
@@ -1,7 +1,4 @@
-import sage.structure.element
-cimport sage.structure.element
-from sage.structure.element cimport RingElement, ModuleElement, CommutativeRingElement
+from sage.structure.element cimport CommutativeRingElement
 
 cdef class LocalGenericElement(CommutativeRingElement):
-    cpdef RingElement _div_(self, RingElement right)
-    cpdef ModuleElement _sub_(self, ModuleElement right)
+    pass
diff --git a/src/sage/rings/padics/local_generic_element.pyx b/src/sage/rings/padics/local_generic_element.pyx
index 6346a139074..9d51e94902a 100644
--- a/src/sage/rings/padics/local_generic_element.pyx
+++ b/src/sage/rings/padics/local_generic_element.pyx
@@ -36,10 +36,10 @@ from sage.structure.element cimport ModuleElement, RingElement, CommutativeRingE
 from sage.structure.element import coerce_binop
 
 cdef class LocalGenericElement(CommutativeRingElement):
-    #cpdef ModuleElement _add_(self, ModuleElement right):
+    #cpdef _add_(self, ModuleElement right):
     #    raise NotImplementedError
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         r"""
         Returns the quotient of ``self`` by ``right``.
 
@@ -369,7 +369,7 @@ cdef class LocalGenericElement(CommutativeRingElement):
     #def __mod__(self, right):
     #    raise NotImplementedError
 
-    #cpdef RingElement _mul_(self, RingElement right):
+    #cpdef _mul_(self, RingElement right):
     #    raise NotImplementedError
 
     #cdef _neg_(self):
@@ -378,7 +378,7 @@ cdef class LocalGenericElement(CommutativeRingElement):
     #def __pow__(self, right):
     #    raise NotImplementedError
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         r"""
         Returns the difference between ``self`` and ``right``.
 
diff --git a/src/sage/rings/padics/padic_ZZ_pX_CA_element.pyx b/src/sage/rings/padics/padic_ZZ_pX_CA_element.pyx
index ac8818ae0f7..6c619a0fbb6 100644
--- a/src/sage/rings/padics/padic_ZZ_pX_CA_element.pyx
+++ b/src/sage/rings/padics/padic_ZZ_pX_CA_element.pyx
@@ -1110,7 +1110,7 @@ cdef class pAdicZZpXCAElement(pAdicZZpXElement):
             return ans
         return self._rshift_c(mpz_get_si((<Integer>shift).value))
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Returns ``-self``.
 
@@ -1377,7 +1377,7 @@ cdef class pAdicZZpXCAElement(pAdicZZpXElement):
         sig_off()
         return ans
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Computes the sum of ``self`` and ``right``.
 
@@ -1416,7 +1416,7 @@ cdef class pAdicZZpXCAElement(pAdicZZpXElement):
             ZZ_pX_add(ans.value, tmpP, right.value)
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Returns the difference of ``self`` and ``right``.
 
@@ -1458,7 +1458,7 @@ cdef class pAdicZZpXCAElement(pAdicZZpXElement):
             ZZ_pX_sub(ans.value, tmpP, right.value)
         return ans
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         Returns the product of ``self`` and ``right``.
 
@@ -1511,7 +1511,7 @@ cdef class pAdicZZpXCAElement(pAdicZZpXElement):
             ZZ_pX_MulMod_pre(ans.value, self_adapted, right_adapted, self.prime_pow.get_modulus_capdiv(ans_absprec)[0])
         return ans
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Returns the quotient of ``self`` by ``right``.
 
diff --git a/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx b/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx
index 7e99cb69809..1fe39db5109 100644
--- a/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx
+++ b/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx
@@ -1766,7 +1766,7 @@ cdef class pAdicZZpXCRElement(pAdicZZpXElement):
                 return ans
         return self._rshift_c(mpz_get_si((<Integer>shift).value))
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Returns ``-self``.
 
@@ -2047,7 +2047,7 @@ cdef class pAdicZZpXCRElement(pAdicZZpXElement):
         sig_off()
         return ans
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Computes the sum of ``self`` and ``right``.
 
@@ -2166,7 +2166,7 @@ cdef class pAdicZZpXCRElement(pAdicZZpXElement):
                 ans.relprec = -ans.relprec
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Returns the difference of ``self`` and ``right``.
 
@@ -2192,7 +2192,7 @@ cdef class pAdicZZpXCRElement(pAdicZZpXElement):
         # For now, a simple implementation
         return self + (-right)
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         Returns the product of ``self`` and ``right``.
 
@@ -2246,7 +2246,7 @@ cdef class pAdicZZpXCRElement(pAdicZZpXElement):
             sig_off()
         return ans
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Returns the quotient of ``self`` by ``right``.
 
diff --git a/src/sage/rings/padics/padic_ZZ_pX_FM_element.pyx b/src/sage/rings/padics/padic_ZZ_pX_FM_element.pyx
index 109d05983dd..e009d781ebc 100644
--- a/src/sage/rings/padics/padic_ZZ_pX_FM_element.pyx
+++ b/src/sage/rings/padics/padic_ZZ_pX_FM_element.pyx
@@ -647,7 +647,7 @@ cdef class pAdicZZpXFMElement(pAdicZZpXElement):
             return ans
         return self._rshift_c(mpz_get_si((<Integer>shift).value))
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Returns ``-self``.
 
@@ -734,7 +734,7 @@ cdef class pAdicZZpXFMElement(pAdicZZpXElement):
             sig_off()
         return ans
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Returns ``self`` + ``right``.
 
@@ -753,7 +753,7 @@ cdef class pAdicZZpXFMElement(pAdicZZpXElement):
         ZZ_pX_add(ans.value, self.value, (<pAdicZZpXFMElement>right).value)
         return ans
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Returns the product of ``self`` and ``right``.
 
@@ -776,7 +776,7 @@ cdef class pAdicZZpXFMElement(pAdicZZpXElement):
         ZZ_pX_MulMod_pre(ans.value, self.value, (<pAdicZZpXFMElement>right).value, self.prime_pow.get_top_modulus()[0])
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Returns the difference of ``self`` and ``right``.
 
@@ -797,7 +797,7 @@ cdef class pAdicZZpXFMElement(pAdicZZpXElement):
         ZZ_pX_sub(ans.value, self.value, (<pAdicZZpXFMElement>right).value)
         return ans
 
-    cpdef RingElement _div_(self, RingElement _right):
+    cpdef _div_(self, RingElement _right):
         """
         Returns the quotient of ``self`` by ``right``.
 
diff --git a/src/sage/rings/padics/padic_generic_element.pxd b/src/sage/rings/padics/padic_generic_element.pxd
index 8def33790f5..5b4641ddf44 100644
--- a/src/sage/rings/padics/padic_generic_element.pxd
+++ b/src/sage/rings/padics/padic_generic_element.pxd
@@ -39,5 +39,3 @@ cdef class pAdicGenericElement(LocalGenericElement):
 
     cpdef abs(self, prec=*)
     cpdef bint _is_base_elt(self, p) except -1
-
-    cpdef RingElement _floordiv_(self, RingElement right)
diff --git a/src/sage/rings/padics/padic_generic_element.pyx b/src/sage/rings/padics/padic_generic_element.pyx
index f1997eafc62..a10c55ceaa5 100644
--- a/src/sage/rings/padics/padic_generic_element.pyx
+++ b/src/sage/rings/padics/padic_generic_element.pyx
@@ -287,7 +287,7 @@ cdef class pAdicGenericElement(LocalGenericElement):
                 raise ZeroDivisionError("cannot divide by zero")
             return self._floordiv_(right)
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Implements floor division.
 
diff --git a/src/sage/rings/polynomial/laurent_polynomial.pyx b/src/sage/rings/polynomial/laurent_polynomial.pyx
index c537e3bb972..f8896df555e 100644
--- a/src/sage/rings/polynomial/laurent_polynomial.pyx
+++ b/src/sage/rings/polynomial/laurent_polynomial.pyx
@@ -496,7 +496,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
                 self.__u = self.__u._parent(coeffs)
         self.__normalize()
 
-    cpdef ModuleElement _add_(self, ModuleElement right_m):
+    cpdef _add_(self, ModuleElement right_m):
         """
         Add two Laurent polynomials with the same parent.
 
@@ -545,7 +545,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         # 3. Add
         return LaurentPolynomial_univariate(self._parent, f1 + f2, m)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right_m):
+    cpdef _sub_(self, ModuleElement right_m):
         """
         Subtract two Laurent polynomials with the same parent.
 
@@ -608,7 +608,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return LaurentPolynomial_univariate(self._parent, -self.__u, self.__n)
 
-    cpdef RingElement _mul_(self, RingElement right_r):
+    cpdef _mul_(self, RingElement right_r):
         """
         EXAMPLES::
 
@@ -623,7 +623,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
                              self.__u * right.__u,
                              self.__n + right.__n)
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         """
         EXAMPLES::
 
@@ -634,7 +634,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return LaurentPolynomial_univariate(self._parent, self.__u._rmul_(c), self.__n)
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         """
         EXAMPLES::
 
@@ -685,7 +685,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
             raise ValueError("exponent must be an integer")
         return LaurentPolynomial_univariate(self._parent, self.__u**right, self.__n*right)
 
-    cpdef RingElement _floordiv_(self, RingElement rhs):
+    cpdef _floordiv_(self, RingElement rhs):
         """
         Perform division with remainder and return the quotient.
 
@@ -755,7 +755,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return LaurentPolynomial_univariate(self._parent, self.__u, self.__n - k)
 
-    cpdef RingElement _div_(self, RingElement rhs):
+    cpdef _div_(self, RingElement rhs):
         """
         EXAMPLES::
 
@@ -1915,7 +1915,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
                 denom *= var[i] ** (-j)
         return (numer, denom)
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         Returns the Laurent polynomial self + right.
 
@@ -1940,7 +1940,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
             ans._poly += right._poly
         return ans
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         Returns the Laurent polynomial self - right.
 
@@ -1967,7 +1967,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
             ans._poly -= right._poly
         return ans
 
-    cpdef RingElement _div_(self, RingElement rhs):
+    cpdef _div_(self, RingElement rhs):
         """
         Return the division of ``self`` by ``rhs``.
 
@@ -2021,7 +2021,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         d = self._poly.dict()
         return len(d) == 1 and 1 in d.values()
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Returns -self.
 
@@ -2038,7 +2038,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         ans._poly = -self._poly
         return ans
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Returns self * right where right is in self's base ring.
 
@@ -2055,7 +2055,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         ans._poly = self._poly * right
         return ans
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Returns left*self where left is in self's base ring.
 
@@ -2072,7 +2072,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         ans._poly = left * self._poly
         return ans
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Return self*right.
 
@@ -2089,7 +2089,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         ans._poly = self._poly * (<LaurentPolynomial_mpair>right)._poly
         return ans
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Perform division with remainder and return the quotient.
 
diff --git a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
index 381a7e1cdb1..e89bbd6c34e 100644
--- a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
+++ b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
@@ -2209,7 +2209,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
         cdef ring *r = (<MPolynomial_libsingular>left)._parent_ring
         return singular_polynomial_cmp(p, q, r)
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         Add left and right.
 
@@ -2225,7 +2225,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
                                  (<MPolynomial_libsingular>right)._poly, r)
         return new_MP((<MPolynomial_libsingular>left)._parent, _p)
 
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         """
         Subtract left and right.
 
@@ -2242,7 +2242,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
                                 _ring)
         return new_MP((<MPolynomial_libsingular>left)._parent, _p)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Multiply self with a base ring element.
 
@@ -2260,7 +2260,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
         singular_polynomial_rmul(&_p, self._poly, left, _ring)
         return new_MP((<MPolynomial_libsingular>self)._parent, _p)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Multiply left and right.
 
@@ -2276,7 +2276,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
         # So, calling _rmul_ is the correct thing to do.
         return self._rmul_(right)
 
-    cpdef RingElement _mul_(left, RingElement right):
+    cpdef _mul_(left, RingElement right):
         """
         Multiply left and right.
 
@@ -2299,7 +2299,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
                                  (<MPolynomial_libsingular>left)._parent_ring)
         return new_MP((<MPolynomial_libsingular>left)._parent,_p)
 
-    cpdef RingElement _div_(left, RingElement right_ringelement):
+    cpdef _div_(left, RingElement right_ringelement):
         """
         Divide left by right
 
@@ -3886,7 +3886,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
         else:
             return False
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Perform division with remainder and return the quotient.
 
diff --git a/src/sage/rings/polynomial/pbori.pyx b/src/sage/rings/polynomial/pbori.pyx
index 625909a08b8..e4a764f5fe9 100644
--- a/src/sage/rings/polynomial/pbori.pyx
+++ b/src/sage/rings/polynomial/pbori.pyx
@@ -2583,7 +2583,7 @@ cdef class BooleanMonomial(MonoidElement):
         """
         return new_BMI_from_BooleanMonomial(self)
 
-    cpdef MonoidElement _mul_(left, MonoidElement right):
+    cpdef _mul_(left, MonoidElement right):
         """
         Multiply this boolean monomial with another boolean monomial.
 
@@ -2969,7 +2969,7 @@ cdef class BooleanPolynomial(MPolynomial):
         R = self.parent().cover_ring()
         return R(self)._latex_()
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         EXAMPLE::
 
@@ -2984,7 +2984,7 @@ cdef class BooleanPolynomial(MPolynomial):
         p._pbpoly.iadd( (<BooleanPolynomial>right)._pbpoly )
         return p
 
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         """
         EXAMPLE::
 
@@ -2996,7 +2996,7 @@ cdef class BooleanPolynomial(MPolynomial):
         """
         return left._add_(right)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLE::
 
@@ -3011,7 +3011,7 @@ cdef class BooleanPolynomial(MPolynomial):
         else:
             return self._parent.zero()
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLE::
 
@@ -3023,7 +3023,7 @@ cdef class BooleanPolynomial(MPolynomial):
         """
         return self._rmul_(right)
 
-    cpdef RingElement _mul_(left, RingElement right):
+    cpdef _mul_(left, RingElement right):
         """
         EXAMPLE::
 
@@ -3038,7 +3038,7 @@ cdef class BooleanPolynomial(MPolynomial):
         p._pbpoly.imul( (<BooleanPolynomial>right)._pbpoly )
         return p
 
-    cpdef RingElement _div_(left, RingElement right):
+    cpdef _div_(left, RingElement right):
         """
         EXAMPLE::
 
diff --git a/src/sage/rings/polynomial/plural.pyx b/src/sage/rings/polynomial/plural.pyx
index adbb8564651..7cfa2f7039e 100644
--- a/src/sage/rings/polynomial/plural.pyx
+++ b/src/sage/rings/polynomial/plural.pyx
@@ -1460,7 +1460,7 @@ cdef class NCPolynomial_plural(RingElement):
         cdef ring *r = (<NCPolynomialRing_plural>left._parent)._ring
         return singular_polynomial_cmp(p, q, r)
 
-    cpdef ModuleElement _add_( left, ModuleElement right):
+    cpdef _add_( left, ModuleElement right):
         """
         Adds left and right.
 
@@ -1479,7 +1479,7 @@ cdef class NCPolynomial_plural(RingElement):
                                  (<NCPolynomialRing_plural>left._parent)._ring)
         return new_NCP((<NCPolynomialRing_plural>left._parent), _p)
 
-    cpdef ModuleElement _sub_( left, ModuleElement right):
+    cpdef _sub_( left, ModuleElement right):
         """
         Subtract left and right.
 
@@ -1501,7 +1501,7 @@ cdef class NCPolynomial_plural(RingElement):
                                 _ring)
         return new_NCP((<NCPolynomialRing_plural>left._parent), _p)
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Multiply ``self`` with a base ring element.
 
@@ -1522,7 +1522,7 @@ cdef class NCPolynomial_plural(RingElement):
         singular_polynomial_rmul(&_p, self._poly, left, _ring)
         return new_NCP((<NCPolynomialRing_plural>self._parent),_p)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Multiply ``self`` with a base ring element.
 
@@ -1568,7 +1568,7 @@ cdef class NCPolynomial_plural(RingElement):
                                  (<NCPolynomialRing_plural>left._parent)._ring)
         return new_NCP((<NCPolynomialRing_plural>left._parent),_p)
 
-    cpdef RingElement _div_(left, RingElement right):
+    cpdef _div_(left, RingElement right):
         """
         Divide left by right
 
diff --git a/src/sage/rings/polynomial/polynomial_element.pxd b/src/sage/rings/polynomial/polynomial_element.pxd
index e35185bd752..8657b0524fe 100644
--- a/src/sage/rings/polynomial/polynomial_element.pxd
+++ b/src/sage/rings/polynomial/polynomial_element.pxd
@@ -5,7 +5,6 @@ from polynomial_compiled import CompiledPolynomialFunction
 from polynomial_compiled cimport CompiledPolynomialFunction
 
 cdef class Polynomial(CommutativeAlgebraElement):
-    cpdef ModuleElement _neg_(self)
     cdef char _is_gen
     cdef CompiledPolynomialFunction _compiled
     cpdef Polynomial truncate(self, long n)
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index 12d568993b5..90b77aedaf8 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -252,7 +252,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
                   v[i] = z
           return v
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         r"""
         Add two polynomials.
 
@@ -281,7 +281,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
         low = [x[i] + y[i] for i in range(min)]
         return self._parent(low + high)
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         return self._parent([-x for x in self.list()])
 
     cpdef bint is_zero(self):
@@ -365,7 +365,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
                 return point(z, *args, **kwds)
         raise NotImplementedError("plotting of polynomials over %s not implemented"%R)
 
-    cpdef ModuleElement _lmul_(self, RingElement left):
+    cpdef _lmul_(self, RingElement left):
         """
         Multiply self on the left by a scalar.
 
@@ -385,7 +385,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
             return self.parent().zero()
         return self.parent()(left) * self
 
-    cpdef ModuleElement _rmul_(self, RingElement right):
+    cpdef _rmul_(self, RingElement right):
         """
         Multiply self on the right by a scalar.
 
@@ -1484,7 +1484,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
             raise TypeError("cannot coerce nonconstant polynomial to long")
         return long(self[0])
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -2365,7 +2365,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
         """
         raise IndexError("polynomials are immutable")
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Quotient of division of self by other. This is denoted //.
 
@@ -8815,7 +8815,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         res.__normalize()
         return res
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         r"""
         Add two polynomials.
 
@@ -8846,7 +8846,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         else:
             return self._new_c(low + high, self._parent)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         cdef Polynomial_generic_dense res
         cdef Py_ssize_t check=0, i, min
         x = (<Polynomial_generic_dense>self).__coeffs
@@ -8867,7 +8867,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         else:
             return self._new_c(low + high, self._parent)
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         if len(self.__coeffs) == 0:
             return self
         if c._parent is not (<Element>self.__coeffs[0])._parent:
@@ -8879,7 +8879,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         res.__normalize()
         return res
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         if len(self.__coeffs) == 0:
             return self
         if c._parent is not (<Element>self.__coeffs[0])._parent:
diff --git a/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx b/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx
index f7fce57bd3e..bc3870ba836 100644
--- a/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx
+++ b/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx
@@ -587,7 +587,7 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
             name = self.parent().latex_variable_names()[0]
         return self._repr(name=name, latex=True)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         r"""
         Returns self plus right.
 
@@ -607,7 +607,7 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
         return x
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         r"""
         Return self minus right.
 
@@ -627,7 +627,7 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
         return x
 
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         r"""
         Returns negative of self.
 
@@ -879,7 +879,7 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
             return self._parent(rr), ss, tt
 
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         r"""
         Returns self multiplied by right.
 
@@ -927,7 +927,7 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
         sig_off()
         return x
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         r"""
         Returns self multiplied by right, where right is a scalar (integer).
 
@@ -946,7 +946,7 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
         return x
 
 
-    cpdef ModuleElement _rmul_(self, RingElement right):
+    cpdef _rmul_(self, RingElement right):
         r"""
         Returns self multiplied by right, where right is a scalar (integer).
 
diff --git a/src/sage/rings/polynomial/polynomial_integer_dense_ntl.pyx b/src/sage/rings/polynomial/polynomial_integer_dense_ntl.pyx
index d32a0580066..ad42320b115 100644
--- a/src/sage/rings/polynomial/polynomial_integer_dense_ntl.pyx
+++ b/src/sage/rings/polynomial/polynomial_integer_dense_ntl.pyx
@@ -422,7 +422,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
             name = self.parent().latex_variable_names()[0]
         return self._repr(name, latex=True)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         r"""
         Returns self plus right.
 
@@ -440,7 +440,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
         return x
 
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         r"""
         Return self minus right.
 
@@ -458,7 +458,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
         return x
 
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         r"""
         Returns negative of self.
 
@@ -664,7 +664,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
             return S(rr), ss, tt
 
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         r"""
         Returns self multiplied by right.
 
@@ -680,7 +680,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
         return x
 
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         r"""
         Returns self multiplied by right, where right is a scalar (integer).
 
@@ -700,7 +700,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
         return x
 
 
-    cpdef ModuleElement _rmul_(self, RingElement right):
+    cpdef _rmul_(self, RingElement right):
         r"""
         Returns self multiplied by right, where right is a scalar (integer).
 
diff --git a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
index 185f2981edd..1f51acb829f 100644
--- a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
+++ b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
@@ -219,10 +219,10 @@ cdef class Polynomial_dense_mod_n(Polynomial):
             return (~self)**(-n)
         return self.parent()(self.__poly**n, construct=True)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         return self.parent()(self.__poly + (<Polynomial_dense_mod_n>right).__poly, construct=True)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -232,13 +232,13 @@ cdef class Polynomial_dense_mod_n(Polynomial):
         """
         return self.parent()(self.__poly * (<Polynomial_dense_mod_n>right).__poly, construct=True)
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         try:
             return self.parent()(ZZ_pX([c], self.parent().modulus()) * self.__poly, construct=True)
         except RuntimeError as msg: # should this really be a TypeError
             raise TypeError(msg)
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         try:
             return self.parent()(ZZ_pX([c], self.parent().modulus()) * self.__poly, construct=True)
         except RuntimeError as msg: # should this really be a TypeError
@@ -289,7 +289,7 @@ cdef class Polynomial_dense_mod_n(Polynomial):
         return self.parent()(self.__poly.left_shift(n),
                              construct=True)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         return self.parent()(self.__poly - (<Polynomial_dense_mod_n>right).__poly, construct=True)
 
     def __floordiv__(self, right):
@@ -675,7 +675,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         self.c.restore_c()
         zz_pX_SetCoeff_long(self.x, n, value)
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         TESTS::
 
@@ -692,7 +692,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         TESTS::
 
@@ -709,7 +709,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         TESTS::
 
@@ -767,7 +767,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         """
         TESTS::
 
@@ -783,7 +783,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         """
         TESTS::
 
@@ -893,7 +893,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         sig_off()
         return q, r
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Returns the whole part of self/right, without remainder.
 
@@ -1226,7 +1226,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         cdef ntl_ZZ_p val = ntl_ZZ_p(a, self.c)
         ZZ_pX_SetCoeff(self.x, n, val.x)
 
-    cpdef ModuleElement _add_(self, ModuleElement _right):
+    cpdef _add_(self, ModuleElement _right):
         """
         TESTS::
 
@@ -1243,7 +1243,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef ModuleElement _sub_(self, ModuleElement _right):
+    cpdef _sub_(self, ModuleElement _right):
         """
         TESTS::
 
@@ -1260,7 +1260,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef RingElement _mul_(self, RingElement _right):
+    cpdef _mul_(self, RingElement _right):
         """
         TESTS::
 
@@ -1318,7 +1318,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         """
         TESTS::
 
@@ -1335,7 +1335,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         if do_sig: sig_off()
         return r
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         """
         TESTS::
 
@@ -1429,7 +1429,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         sig_off()
         return q, r
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Returns the whole part of self/right, without remainder.
 
diff --git a/src/sage/rings/polynomial/polynomial_rational_flint.pyx b/src/sage/rings/polynomial/polynomial_rational_flint.pyx
index 5381125e0b2..a04064aba6a 100644
--- a/src/sage/rings/polynomial/polynomial_rational_flint.pyx
+++ b/src/sage/rings/polynomial/polynomial_rational_flint.pyx
@@ -778,7 +778,7 @@ cdef class Polynomial_rational_flint(Polynomial):
     # Arithmetic                                                              #
     ###########################################################################
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Returns the sum of two rational polynomials.
 
@@ -806,7 +806,7 @@ cdef class Polynomial_rational_flint(Polynomial):
         if do_sig: sig_off()
         return res
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Returns the difference of two rational polynomials.
 
@@ -834,7 +834,7 @@ cdef class Polynomial_rational_flint(Polynomial):
         if do_sig: sig_off()
         return res
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Returns the difference of two rational polynomials.
 
@@ -988,7 +988,7 @@ cdef class Polynomial_rational_flint(Polynomial):
         sig_off()
         return d, s, t
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Returns the product of self and right.
 
@@ -1054,7 +1054,7 @@ cdef class Polynomial_rational_flint(Polynomial):
         if do_sig: sig_off()
         return res
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         r"""
         Returns left * self, where left is a rational number.
 
@@ -1074,7 +1074,7 @@ cdef class Polynomial_rational_flint(Polynomial):
         if do_sig: sig_off()
         return res
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         r"""
         Returns self * right, where right is a rational number.
 
diff --git a/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx b/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx
index 3eefe39c229..67a27853302 100644
--- a/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx
+++ b/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx
@@ -378,7 +378,7 @@ cdef class PolynomialRealDense(Polynomial):
             mpfr_neg(f._coeffs[i], self._coeffs[i], rnd)
         return f
 
-    cpdef ModuleElement _add_(left, ModuleElement _right):
+    cpdef _add_(left, ModuleElement _right):
         """
         EXAMPLES::
 
@@ -411,7 +411,7 @@ cdef class PolynomialRealDense(Polynomial):
         f._normalize()
         return f
 
-    cpdef ModuleElement _sub_(left, ModuleElement _right):
+    cpdef _sub_(left, ModuleElement _right):
         """
         EXAMPLES::
 
@@ -442,10 +442,10 @@ cdef class PolynomialRealDense(Polynomial):
         f._normalize()
         return f
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         return self._lmul_(c)
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         """
         EXAMPLES::
 
@@ -467,7 +467,7 @@ cdef class PolynomialRealDense(Polynomial):
             mpfr_mul(f._coeffs[i], self._coeffs[i], a.value, rnd)
         return f
 
-    cpdef RingElement _mul_(left, RingElement _right):
+    cpdef _mul_(left, RingElement _right):
         """
         Here we use the naive `O(n^2)` algorithm, as asymptotically faster algorithms such
         as Karatsuba can have very inaccurate results due to intermediate rounding errors.
diff --git a/src/sage/rings/polynomial/polynomial_template.pxi b/src/sage/rings/polynomial/polynomial_template.pxi
index 436df32923c..e71d9cb0f51 100644
--- a/src/sage/rings/polynomial/polynomial_template.pxi
+++ b/src/sage/rings/polynomial/polynomial_template.pxi
@@ -222,7 +222,7 @@ cdef class Polynomial_template(Polynomial):
         """
         celement_destruct(&self.x, (<Polynomial_template>self)._cparent)
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         EXAMPLE::
 
@@ -240,7 +240,7 @@ cdef class Polynomial_template(Polynomial):
         #assert(r._parent(pari(self) + pari(right)) == r)
         return r
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         EXAMPLE::
 
@@ -274,7 +274,7 @@ cdef class Polynomial_template(Polynomial):
         #assert(r._parent(-pari(self)) == r)
         return r
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLES::
 
@@ -304,7 +304,7 @@ cdef class Polynomial_template(Polynomial):
         celement_mul_scalar(&r.x, &(<Polynomial_template>self).x, left, (<Polynomial_template>self)._cparent)
         return r
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         EXAMPLES::
 
@@ -325,7 +325,7 @@ cdef class Polynomial_template(Polynomial):
         # all currently implemented rings are commutative
         return self._rmul_(right)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         EXAMPLE::
 
@@ -412,7 +412,7 @@ cdef class Polynomial_template(Polynomial):
         #assert(t._parent(tp) == t)
         return r,s,t
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         EXAMPLES::
 
diff --git a/src/sage/rings/polynomial/polynomial_zmod_flint.pxd b/src/sage/rings/polynomial/polynomial_zmod_flint.pxd
index 9809d8f0813..d8cef40282f 100644
--- a/src/sage/rings/polynomial/polynomial_zmod_flint.pxd
+++ b/src/sage/rings/polynomial/polynomial_zmod_flint.pxd
@@ -13,5 +13,5 @@ cdef class Polynomial_zmod_flint(Polynomial_template):
     cdef Polynomial_template _new(self)
     cdef int _set_list(self, x) except -1
     cdef int _set_fmpz_poly(self, fmpz_poly_t) except -1
-    cpdef _mul_trunc_opposite(self, Polynomial_zmod_flint other, length)
+    cpdef Polynomial _mul_trunc_opposite(self, Polynomial_zmod_flint other, length)
     cpdef rational_reconstruct(self, m, n_deg=?, d_deg=?)
diff --git a/src/sage/rings/polynomial/polynomial_zmod_flint.pyx b/src/sage/rings/polynomial/polynomial_zmod_flint.pyx
index e2d8532ea97..ffc62fd8a61 100644
--- a/src/sage/rings/polynomial/polynomial_zmod_flint.pyx
+++ b/src/sage/rings/polynomial/polynomial_zmod_flint.pyx
@@ -496,7 +496,7 @@ cdef class Polynomial_zmod_flint(Polynomial_template):
 
     _mul_short = _mul_trunc_
 
-    cpdef _mul_trunc_opposite(self, Polynomial_zmod_flint other, n):
+    cpdef Polynomial _mul_trunc_opposite(self, Polynomial_zmod_flint other, n):
         """
         Return the product of this polynomial and other ignoring the least
         significant `n` terms of the result which may be set to anything.
diff --git a/src/sage/rings/polynomial/polynomial_zz_pex.pyx b/src/sage/rings/polynomial/polynomial_zz_pex.pyx
index dabff28e310..d863fcb9925 100644
--- a/src/sage/rings/polynomial/polynomial_zz_pex.pyx
+++ b/src/sage/rings/polynomial/polynomial_zz_pex.pyx
@@ -193,7 +193,7 @@ cdef class Polynomial_ZZ_pEX(Polynomial_template):
         K = self._parent.base_ring()
         return [K(ZZ_pE_c_to_list(ZZ_pEX_coeff(self.x, i))) for i in range(celement_len(&self.x, (<Polynomial_template>self)._cparent))]
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         EXAMPLE::
             sage: K.<a>=GF(next_prime(2**60)**3)
diff --git a/src/sage/rings/power_series_mpoly.pyx b/src/sage/rings/power_series_mpoly.pyx
index 6642a0f515c..70dcb373541 100644
--- a/src/sage/rings/power_series_mpoly.pyx
+++ b/src/sage/rings/power_series_mpoly.pyx
@@ -103,7 +103,7 @@ cdef class PowerSeries_mpoly(PowerSeries):
     def _mpoly(self):
         return self.__f
 
-    cpdef RingElement _mul_(self, RingElement right_r):
+    cpdef _mul_(self, RingElement right_r):
         """
         Return the product of two power series.
         """
@@ -127,7 +127,7 @@ cdef class PowerSeries_mpoly(PowerSeries):
         return PowerSeries_mpoly(self._parent, -self.__f,
                                          self._prec, check=False)
 
-    cpdef ModuleElement _add_(self, ModuleElement right_m):
+    cpdef _add_(self, ModuleElement right_m):
         """
         EXAMPLES:
         """
@@ -135,7 +135,7 @@ cdef class PowerSeries_mpoly(PowerSeries):
         return PowerSeries_mpoly(self._parent, self.__f + right.__f, \
                                          self.common_prec_c(right), check=True)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right_m):
+    cpdef _sub_(self, ModuleElement right_m):
         """
         Return difference of two power series.
 
@@ -145,10 +145,10 @@ cdef class PowerSeries_mpoly(PowerSeries):
         return PowerSeries_mpoly(self._parent, self.__f - right.__f, \
                                          self.common_prec_c(right), check=True)
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         return PowerSeries_mpoly(self._parent, self.__f._rmul_(c), self._prec, check=False)
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         return PowerSeries_mpoly(self._parent, self.__f._lmul_(c), self._prec, check=False)
 
 
diff --git a/src/sage/rings/power_series_poly.pyx b/src/sage/rings/power_series_poly.pyx
index 9ef90db4695..6b1705df6b5 100644
--- a/src/sage/rings/power_series_poly.pyx
+++ b/src/sage/rings/power_series_poly.pyx
@@ -467,7 +467,7 @@ cdef class PowerSeries_poly(PowerSeries):
         return PowerSeries_poly(self._parent, -self.__f,
                                          self._prec, check=False)
 
-    cpdef ModuleElement _add_(self, ModuleElement right_m):
+    cpdef _add_(self, ModuleElement right_m):
         """
         EXAMPLES::
 
@@ -498,7 +498,7 @@ cdef class PowerSeries_poly(PowerSeries):
         return PowerSeries_poly(self._parent, self.__f + right.__f, \
                                          self.common_prec_c(right), check=True)
 
-    cpdef ModuleElement _sub_(self, ModuleElement right_m):
+    cpdef _sub_(self, ModuleElement right_m):
         """
         Return the difference of two power series.
 
@@ -513,7 +513,7 @@ cdef class PowerSeries_poly(PowerSeries):
         return PowerSeries_poly(self._parent, self.__f - right.__f, \
                                          self.common_prec_c(right), check=True)
 
-    cpdef RingElement _mul_(self, RingElement right_r):
+    cpdef _mul_(self, RingElement right_r):
         """
         Return the product of two power series.
 
@@ -529,7 +529,7 @@ cdef class PowerSeries_poly(PowerSeries):
                                 prec = prec,
                                 check = True)  # check, since truncation may be needed
 
-    cpdef ModuleElement _rmul_(self, RingElement c):
+    cpdef _rmul_(self, RingElement c):
         """
         Multiply self on the right by a scalar.
 
@@ -542,7 +542,7 @@ cdef class PowerSeries_poly(PowerSeries):
         """
         return PowerSeries_poly(self._parent, self.__f * c, self._prec, check=False)
 
-    cpdef ModuleElement _lmul_(self, RingElement c):
+    cpdef _lmul_(self, RingElement c):
         """
         Multiply self on the left by a scalar.
 
diff --git a/src/sage/rings/power_series_ring_element.pyx b/src/sage/rings/power_series_ring_element.pyx
index 2c891df0e2b..63a90ecf1dc 100644
--- a/src/sage/rings/power_series_ring_element.pyx
+++ b/src/sage/rings/power_series_ring_element.pyx
@@ -1005,7 +1005,7 @@ cdef class PowerSeries(AlgebraElement):
                     v[k-n] = x
         return self._parent(v, self.prec()-n)
 
-    cpdef RingElement _div_(self, RingElement denom_r):
+    cpdef _div_(self, RingElement denom_r):
         """
         EXAMPLES::
 
@@ -1039,7 +1039,7 @@ cdef class PowerSeries(AlgebraElement):
             num = self
         return num*inv
 
-    cpdef RingElement _floordiv_(self, RingElement denom):
+    cpdef _floordiv_(self, RingElement denom):
         """
         Euclidean division (over fields) or ordinary division (over
         other rings; deprecated).
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index 4cd4ec1fc96..b01193bd656 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -2076,7 +2076,7 @@ cdef class Rational(sage.structure.element.FieldElement):
     ################################################################
     # Optimized arithmetic
     ################################################################
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Return ``right`` plus ``self``.
 
@@ -2092,7 +2092,7 @@ cdef class Rational(sage.structure.element.FieldElement):
         mpq_add(x.value, self.value, (<Rational>right).value)
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Return ``self`` minus ``right``.
 
@@ -2107,7 +2107,7 @@ cdef class Rational(sage.structure.element.FieldElement):
         mpq_sub(x.value, self.value, (<Rational>right).value)
         return x
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Negate ``self``.
 
@@ -2121,7 +2121,7 @@ cdef class Rational(sage.structure.element.FieldElement):
         mpq_neg(x.value, self.value)
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Return ``self`` times ``right``.
 
@@ -2143,7 +2143,7 @@ cdef class Rational(sage.structure.element.FieldElement):
             mpq_mul(x.value, self.value, (<Rational>right).value)
         return x
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Return ``self`` divided by ``right``.
 
diff --git a/src/sage/rings/real_arb.pyx b/src/sage/rings/real_arb.pyx
index f97557eab33..acb9375c6da 100644
--- a/src/sage/rings/real_arb.pyx
+++ b/src/sage/rings/real_arb.pyx
@@ -2610,7 +2610,7 @@ cdef class RealBall(RingElement):
         if _do_sig(prec(self)): sig_off()
         return res
 
-    cpdef ModuleElement _add_(self, ModuleElement other):
+    cpdef _add_(self, ModuleElement other):
         """
         Return the sum of two balls, rounded to the ambient field's precision.
 
@@ -2628,7 +2628,7 @@ cdef class RealBall(RingElement):
         if _do_sig(prec(self)): sig_off()
         return res
 
-    cpdef ModuleElement _sub_(self, ModuleElement other):
+    cpdef _sub_(self, ModuleElement other):
         """
         Return the difference of two balls, rounded to the ambient field's
         precision.
@@ -2647,7 +2647,7 @@ cdef class RealBall(RingElement):
         if _do_sig(prec(self)): sig_off()
         return res
 
-    cpdef RingElement _mul_(self, RingElement other):
+    cpdef _mul_(self, RingElement other):
         """
         Return the product of two balls, rounded to the ambient field's
         precision.
@@ -2666,7 +2666,7 @@ cdef class RealBall(RingElement):
         if _do_sig(prec(self)): sig_off()
         return res
 
-    cpdef RingElement _div_(self, RingElement other):
+    cpdef _div_(self, RingElement other):
         """
         Return the quotient of two balls, rounded to the ambient field's
         precision.
diff --git a/src/sage/rings/real_double.pyx b/src/sage/rings/real_double.pyx
index 8add1808723..21334f557e6 100644
--- a/src/sage/rings/real_double.pyx
+++ b/src/sage/rings/real_double.pyx
@@ -1218,7 +1218,7 @@ cdef class RealDoubleElement(FieldElement):
         x._value = 1.0 / self._value
         return x
 
-    cpdef ModuleElement _add_(self, ModuleElement right):
+    cpdef _add_(self, ModuleElement right):
         """
         Add two real numbers with the same parent.
 
@@ -1231,7 +1231,7 @@ cdef class RealDoubleElement(FieldElement):
         x._value = self._value + (<RealDoubleElement>right)._value
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two real numbers with the same parent.
 
@@ -1244,7 +1244,7 @@ cdef class RealDoubleElement(FieldElement):
         x._value = self._value - (<RealDoubleElement>right)._value
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply two real numbers with the same parent.
 
@@ -1257,7 +1257,7 @@ cdef class RealDoubleElement(FieldElement):
         x._value = self._value * (<RealDoubleElement>right)._value
         return x
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide ``self`` by ``right``.
 
diff --git a/src/sage/rings/real_interval_absolute.pyx b/src/sage/rings/real_interval_absolute.pyx
index a319194da53..da2749191be 100644
--- a/src/sage/rings/real_interval_absolute.pyx
+++ b/src/sage/rings/real_interval_absolute.pyx
@@ -634,7 +634,7 @@ cdef class RealIntervalAbsoluteElement(FieldElement):
         else:
             return self._new_c(zero, max(-self._mantissa, self._mantissa + self._diameter))
 
-    cpdef ModuleElement _add_(self, ModuleElement _other):
+    cpdef _add_(self, ModuleElement _other):
         """
         TESTS::
 
@@ -652,7 +652,7 @@ cdef class RealIntervalAbsoluteElement(FieldElement):
         cdef RealIntervalAbsoluteElement other = <RealIntervalAbsoluteElement>_other
         return self._new_c(self._mantissa + other._mantissa, self._diameter + other._diameter)
 
-    cpdef ModuleElement _sub_(self, ModuleElement _other):
+    cpdef _sub_(self, ModuleElement _other):
         """
         TESTS::
 
@@ -672,7 +672,7 @@ cdef class RealIntervalAbsoluteElement(FieldElement):
         cdef RealIntervalAbsoluteElement other = <RealIntervalAbsoluteElement>_other
         return self._new_c(self._mantissa - other._mantissa - other._diameter, self._diameter + other._diameter)
 
-    cpdef RingElement _mul_(self, RingElement _other):
+    cpdef _mul_(self, RingElement _other):
         """
         TESTS::
 
@@ -832,7 +832,7 @@ cdef class RealIntervalAbsoluteElement(FieldElement):
             res = -res
         return res
 
-    cpdef RingElement _div_(self, RingElement _other):
+    cpdef _div_(self, RingElement _other):
         """
         TESTS::
 
diff --git a/src/sage/rings/real_lazy.pyx b/src/sage/rings/real_lazy.pyx
index 2e9c2a5b4c2..42a4a6878cf 100644
--- a/src/sage/rings/real_lazy.pyx
+++ b/src/sage/rings/real_lazy.pyx
@@ -545,7 +545,7 @@ cdef int get_new_prec(R, int depth) except -1:
 
 cdef class LazyFieldElement(FieldElement):
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         Add ``left`` with ``right``.
 
@@ -561,7 +561,7 @@ cdef class LazyFieldElement(FieldElement):
                 pass
         return left._new_binop(left, right, add)
 
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         """
         Subtract ``right`` from ``left``.
 
@@ -577,7 +577,7 @@ cdef class LazyFieldElement(FieldElement):
                 pass
         return left._new_binop(left, right, sub)
 
-    cpdef RingElement _mul_(left, RingElement right):
+    cpdef _mul_(left, RingElement right):
         """
         Mutliply ``left`` with ``right``.
 
@@ -593,7 +593,7 @@ cdef class LazyFieldElement(FieldElement):
                 pass
         return left._new_binop(left, right, mul)
 
-    cpdef RingElement _div_(left, RingElement right):
+    cpdef _div_(left, RingElement right):
         """
         Divide ``left`` by ``right``.
 
diff --git a/src/sage/rings/real_mpfi.pyx b/src/sage/rings/real_mpfi.pyx
index d07a54b29ad..69a98b4c39d 100644
--- a/src/sage/rings/real_mpfi.pyx
+++ b/src/sage/rings/real_mpfi.pyx
@@ -2562,7 +2562,7 @@ cdef class RealIntervalFieldElement(RingElement):
     #   Basic Arithmetic
     ########################
 
-    cpdef ModuleElement _add_(self, ModuleElement other):
+    cpdef _add_(self, ModuleElement other):
         """
         Add two real intervals with the same parent.
 
@@ -2608,7 +2608,7 @@ cdef class RealIntervalFieldElement(RingElement):
         mpfi_inv(x.value, self.value)
         return x
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two real intervals with the same parent.
 
@@ -2627,7 +2627,7 @@ cdef class RealIntervalFieldElement(RingElement):
         mpfi_sub(x.value, self.value, (<RealIntervalFieldElement>right).value)
         return x
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply two real intervals with the same parent.
 
@@ -2665,7 +2665,7 @@ cdef class RealIntervalFieldElement(RingElement):
         return x
 
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide ``self`` by ``right``, where both are real intervals with the
         same parent.
@@ -2693,7 +2693,7 @@ cdef class RealIntervalFieldElement(RingElement):
                  (<RealIntervalFieldElement>right).value)
         return x
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Return the additive "inverse" of this interval. (Technically,
         non-precise intervals don't have additive inverses.)
diff --git a/src/sage/rings/real_mpfr.pyx b/src/sage/rings/real_mpfr.pyx
index f27a21c76ac..88b56987c61 100644
--- a/src/sage/rings/real_mpfr.pyx
+++ b/src/sage/rings/real_mpfr.pyx
@@ -2194,7 +2194,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
     #   Basic Arithmetic
     ########################
 
-    cpdef ModuleElement _add_(self, ModuleElement other):
+    cpdef _add_(self, ModuleElement other):
         """
         Add two real numbers with the same parent.
 
@@ -2223,7 +2223,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
         """
         return self._parent(1) / self
 
-    cpdef ModuleElement _sub_(self, ModuleElement right):
+    cpdef _sub_(self, ModuleElement right):
         """
         Subtract two real numbers with the same parent.
 
@@ -2253,7 +2253,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
         import sympy
         return sympy.simplify(float(self))
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Multiply two real numbers with the same parent.
 
@@ -2286,7 +2286,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
         return x
 
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Divide ``self`` by other, where both are real numbers with the same
         parent.
@@ -2306,7 +2306,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
                  (<RealNumber>right).value, (<RealField_class>self._parent).rnd)
         return x
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         """
         Return the negative of ``self``.
 
diff --git a/src/sage/rings/semirings/tropical_semiring.pyx b/src/sage/rings/semirings/tropical_semiring.pyx
index fa06154e1e6..4ab47ab06d2 100644
--- a/src/sage/rings/semirings/tropical_semiring.pyx
+++ b/src/sage/rings/semirings/tropical_semiring.pyx
@@ -210,7 +210,7 @@ cdef class TropicalSemiringElement(RingElement):
             return 1
         return 0
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         Add ``left`` to ``right``.
 
@@ -264,7 +264,7 @@ cdef class TropicalSemiringElement(RingElement):
             return self
         raise ArithmeticError("cannot negate any non-infinite element")
 
-    cpdef RingElement _mul_(left, RingElement right):
+    cpdef _mul_(left, RingElement right):
         """
         Multiply ``left`` and ``right``.
 
@@ -290,7 +290,7 @@ cdef class TropicalSemiringElement(RingElement):
         x._val = self._val + rhs._val
         return x
 
-    cpdef RingElement _div_(left, RingElement right):
+    cpdef _div_(left, RingElement right):
         """
         Divide ``left`` by ``right``.
 
diff --git a/src/sage/schemes/toric/divisor_class.pyx b/src/sage/schemes/toric/divisor_class.pyx
index d6b355eaa69..aa1c95c8aa1 100644
--- a/src/sage/schemes/toric/divisor_class.pyx
+++ b/src/sage/schemes/toric/divisor_class.pyx
@@ -209,7 +209,7 @@ cdef class ToricRationalDivisorClass(Vector_rational_dense):
         # Now let the standard framework work...
         return Vector_rational_dense._act_on_(self, other, self_on_left)
 
-    cpdef Element _dot_product_(self, Vector right):
+    cpdef _dot_product_(self, Vector right):
         r"""
         Raise a ``TypeError`` exception.
 
diff --git a/src/sage/structure/element.pxd b/src/sage/structure/element.pxd
index 87b586c7d80..95b16ef7d41 100644
--- a/src/sage/structure/element.pxd
+++ b/src/sage/structure/element.pxd
@@ -50,36 +50,34 @@ cdef class ModuleElement(Element)       # forward declaration
 cdef class RingElement(ModuleElement)   # forward declaration
 
 cdef class ModuleElement(Element):
-
-    cpdef ModuleElement _add_(self, ModuleElement right)
-    cpdef ModuleElement _sub_(self, ModuleElement right)
-    cpdef ModuleElement _neg_(self)
+    cpdef _add_(self, ModuleElement right)
+    cpdef _sub_(self, ModuleElement right)
+    cpdef _neg_(self)
     # self._rmul_(x) is x * self
-    cpdef ModuleElement _lmul_(self, RingElement right)
+    cpdef _lmul_(self, RingElement right)
     # self._lmul_(x) is self * x, to abide with Python conventions.
-    cpdef ModuleElement _rmul_(self, RingElement left)
+    cpdef _rmul_(self, RingElement left)
 
-    cdef ModuleElement _mul_long(self, long n)
+    cdef _mul_long(self, long n)
 
     # Coerce x to the base ring of self and return the result.
     cdef RingElement coerce_to_base_ring(self, x)
 
 cdef class MonoidElement(Element):
-    cpdef MonoidElement _mul_(self, MonoidElement right)
+    cpdef _mul_(self, MonoidElement right)
 
 cdef class MultiplicativeGroupElement(MonoidElement):
-    cpdef MultiplicativeGroupElement _div_(self, MultiplicativeGroupElement right)
-
+    cpdef _div_(self, MultiplicativeGroupElement right)
 
 cdef class AdditiveGroupElement(ModuleElement):
     pass
 
 cdef class RingElement(ModuleElement):
-    cpdef RingElement _mul_(self, RingElement right)
-    cpdef RingElement _div_(self, RingElement right)
-    cpdef RingElement _floordiv_(self, RingElement right)
+    cpdef _mul_(self, RingElement right)
+    cpdef _div_(self, RingElement right)
+    cpdef _floordiv_(self, RingElement right)
 
-    cdef RingElement _add_long(self, long n)
+    cdef _add_long(self, long n)
 
 cdef class CommutativeRingElement(RingElement):
     pass
@@ -118,10 +116,10 @@ cdef class Vector(ModuleElement):
     # Return the dot product using the simple metric
     # $e_i \cdot e_j = \delta_{ij}$. The first assumes that the parents
     # are equal, both assume that the degrees are equal.
-    cpdef Element _dot_product_(Vector left, Vector right)
-    cpdef Element _dot_product_coerce_(Vector left, Vector right)
+    cpdef _dot_product_(Vector left, Vector right)
+    cpdef _dot_product_coerce_(Vector left, Vector right)
 
-    cpdef Vector _pairwise_product_(Vector left, Vector right) # override, call if parents the same
+    cpdef _pairwise_product_(Vector left, Vector right) # override, call if parents the same
 
     cdef bint is_sparse_c(self)
     cdef bint is_dense_c(self)
@@ -132,9 +130,9 @@ cdef class Matrix(ModuleElement):
     cdef Py_ssize_t _nrows
     cdef Py_ssize_t _ncols
 
-    cdef Vector _vector_times_matrix_(matrix_right, Vector vector_left)    # OK to override, AND call directly
-    cdef Vector _matrix_times_vector_(matrix_left, Vector vector_right)    # OK to override, AND call directly
-    cdef Matrix _matrix_times_matrix_(left, Matrix right)                  # OK to override, AND call directly
+    cdef _vector_times_matrix_(matrix_right, Vector vector_left)    # OK to override, AND call directly
+    cdef _matrix_times_vector_(matrix_left, Vector vector_right)    # OK to override, AND call directly
+    cdef _matrix_times_matrix_(left, Matrix right)                  # OK to override, AND call directly
 
     cdef bint is_sparse_c(self)
     cdef bint is_dense_c(self)
diff --git a/src/sage/structure/element.pyx b/src/sage/structure/element.pyx
index 2e7dea3a8c8..8e7f5bed4b7 100644
--- a/src/sage/structure/element.pyx
+++ b/src/sage/structure/element.pyx
@@ -1308,7 +1308,7 @@ cdef class ModuleElement(Element):
             return (<ModuleElement>left)._add_(<ModuleElement>right)
         return coercion_model.bin_op(left, right, add)
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         raise TypeError(arith_error_message(left, right, add))
 
     def __iadd__(ModuleElement self, right):
@@ -1329,7 +1329,7 @@ cdef class ModuleElement(Element):
             return (<ModuleElement>left)._sub_(<ModuleElement>right)
         return coercion_model.bin_op(left, right, sub)
 
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         # default implementation is to use the negation and addition
         # dispatchers:
         return left._add_(-right)
@@ -1351,7 +1351,7 @@ cdef class ModuleElement(Element):
         """
         return self._neg_()
 
-    cpdef ModuleElement _neg_(self):
+    cpdef _neg_(self):
         # default implementation is to try multiplying by -1.
         if self._parent._base is None:
             return coercion_model.bin_op(-1, self, mul)
@@ -1376,7 +1376,7 @@ cdef class ModuleElement(Element):
         return coercion_model.bin_op(left, right, imul)
 
     # rmul -- left * self
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         """
         Default module left scalar multiplication, which is to try to
         canonically coerce the scalar to the integers and do that
@@ -1388,7 +1388,7 @@ cdef class ModuleElement(Element):
 
     # lmul -- self * right
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Default module left scalar multiplication, which is to try to
         canonically coerce the scalar to the integers and do that
@@ -1398,7 +1398,7 @@ cdef class ModuleElement(Element):
         """
         return None
 
-    cdef ModuleElement _mul_long(self, long n):
+    cdef _mul_long(self, long n):
         """
         Generic path for multiplying by a C long, assumed to commute.
         """
@@ -1462,7 +1462,7 @@ cdef class MonoidElement(Element):
             raise
 
 
-    cpdef MonoidElement _mul_(left, MonoidElement right):
+    cpdef _mul_(left, MonoidElement right):
         """
         Cython classes should override this function to implement multiplication.
         See extensive documentation at the top of element.pyx.
@@ -1537,10 +1537,10 @@ cdef class AdditiveGroupElement(ModuleElement):
     def __invert__(self):
         raise NotImplementedError("multiplicative inverse not defined for additive group elements")
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         return self._lmul_(left)
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         """
         Default module left scalar multiplication, which is to try to
         canonically coerce the scalar to the integers and do that
@@ -1598,7 +1598,7 @@ cdef class MultiplicativeGroupElement(MonoidElement):
             return (<MultiplicativeGroupElement>left)._div_(<MultiplicativeGroupElement>right)
         return coercion_model.bin_op(left, right, div)
 
-    cpdef MultiplicativeGroupElement _div_(self, MultiplicativeGroupElement right):
+    cpdef _div_(self, MultiplicativeGroupElement right):
         """
         Cython classes should override this function to implement division.
         See extensive documentation at the top of element.pyx.
@@ -1643,7 +1643,7 @@ cdef class RingElement(ModuleElement):
             return (<RingElement>right)._add_long(PyInt_AS_LONG(left))
         return coercion_model.bin_op(left, right, add)
 
-    cdef RingElement _add_long(self, long n):
+    cdef _add_long(self, long n):
         """
         Generic path for adding a C long, assumed to commute.
         """
@@ -1669,11 +1669,11 @@ cdef class RingElement(ModuleElement):
     # Multiplication
     ##################################
 
-    cpdef ModuleElement _lmul_(self, RingElement right):
+    cpdef _lmul_(self, RingElement right):
         # We return None to invoke the default action of coercing into self
         return None
 
-    cpdef ModuleElement _rmul_(self, RingElement left):
+    cpdef _rmul_(self, RingElement left):
         # We return None to invoke the default action of coercing into self
         return None
 
@@ -1797,7 +1797,7 @@ cdef class RingElement(ModuleElement):
             return (<ModuleElement>right)._mul_long(PyInt_AS_LONG(left))
         return coercion_model.bin_op(left, right, mul)
 
-    cpdef RingElement _mul_(self, RingElement right):
+    cpdef _mul_(self, RingElement right):
         """
         Cython classes should override this function to implement multiplication.
         See extensive documentation at the top of element.pyx.
@@ -1951,7 +1951,7 @@ cdef class RingElement(ModuleElement):
             return (<RingElement>self)._div_(<RingElement>right)
         return coercion_model.bin_op(self, right, div)
 
-    cpdef RingElement _div_(self, RingElement right):
+    cpdef _div_(self, RingElement right):
         """
         Cython classes should override this function to implement division.
         See extensive documentation at the top of element.pyx.
@@ -1997,7 +1997,7 @@ cdef class RingElement(ModuleElement):
             return (<RingElement>self)._floordiv_(<RingElement>right)
         return coercion_model.bin_op(self, right, floordiv)
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Cython classes should override this function to implement floor
         division. See extensive documentation at the top of element.pyx.
@@ -2697,13 +2697,13 @@ cdef class Vector(ModuleElement):
             return (<Vector>left)._dot_product_(<Vector>right)
         return coercion_model.bin_op(left, right, mul)
 
-    cpdef Element _dot_product_(Vector left, Vector right):
+    cpdef _dot_product_(Vector left, Vector right):
         return left._dot_product_coerce_(right)
 
-    cpdef Element _dot_product_coerce_(Vector left, Vector right):
+    cpdef _dot_product_coerce_(Vector left, Vector right):
         raise TypeError(arith_error_message(left, right, mul))
 
-    cpdef Vector _pairwise_product_(Vector left, Vector right):
+    cpdef _pairwise_product_(Vector left, Vector right):
         raise TypeError("unsupported operation for '%s' and '%s'"%(parent_c(left), parent_c(right)))
 
     def __truediv__(self, right):
@@ -3042,13 +3042,13 @@ cdef class Matrix(ModuleElement):
             return left * ~right
         return coercion_model.bin_op(left, right, div)
 
-    cdef Vector _vector_times_matrix_(matrix_right, Vector vector_left):
+    cdef _vector_times_matrix_(matrix_right, Vector vector_left):
         raise TypeError
 
-    cdef Vector _matrix_times_vector_(matrix_left, Vector vector_right):
+    cdef _matrix_times_vector_(matrix_left, Vector vector_right):
         raise TypeError
 
-    cdef Matrix _matrix_times_matrix_(left, Matrix right):
+    cdef _matrix_times_matrix_(left, Matrix right):
         raise TypeError
 
 
@@ -3135,7 +3135,7 @@ cdef class EuclideanDomainElement(PrincipalIdealDomainElement):
             x, y = canonical_coercion(self, other)
             return x.quo_rem(y)
 
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Quotient of division of ``self`` by other.  This is denoted //.
 
@@ -3194,7 +3194,7 @@ def is_FieldElement(x):
     return isinstance(x, FieldElement)
 
 cdef class FieldElement(CommutativeRingElement):
-    cpdef RingElement _floordiv_(self, RingElement right):
+    cpdef _floordiv_(self, RingElement right):
         """
         Return the quotient of self and other. Since these are field
         elements, the floor division is exactly the same as usual division.
diff --git a/src/sage/symbolic/expression.pyx b/src/sage/symbolic/expression.pyx
index 3d53cac55f4..2fd32e87116 100644
--- a/src/sage/symbolic/expression.pyx
+++ b/src/sage/symbolic/expression.pyx
@@ -2822,7 +2822,7 @@ cdef class Expression(CommutativeRingElement):
         except TypeError:
             return self._parent._coerce_(z)
 
-    cpdef ModuleElement _add_(left, ModuleElement right):
+    cpdef _add_(left, ModuleElement right):
         """
         Add left and right.
 
@@ -2917,7 +2917,7 @@ cdef class Expression(CommutativeRingElement):
             x = gadd(left._gobj, _right._gobj)
         return new_Expression_from_GEx(left._parent, x)
 
-    cpdef ModuleElement _sub_(left, ModuleElement right):
+    cpdef _sub_(left, ModuleElement right):
         """
         EXAMPLES::
 
@@ -2969,7 +2969,7 @@ cdef class Expression(CommutativeRingElement):
             x = gsub(left._gobj, _right._gobj)
         return new_Expression_from_GEx(left._parent, x)
 
-    cpdef RingElement _mul_(left, RingElement right):
+    cpdef _mul_(left, RingElement right):
         """
         Multiply left and right.
 
@@ -3202,7 +3202,7 @@ cdef class Expression(CommutativeRingElement):
             x = gmul(left._gobj, _right._gobj)
         return new_Expression_from_GEx(left._parent, x)
 
-    cpdef RingElement _div_(left, RingElement right):
+    cpdef _div_(left, RingElement right):
         """
         Divide left and right.