sagemathgh-37316: use CommutativeRing in ring_extension

    
Removing one of the last uses of `CommutativeAlgebra` class, towards its
deprecation.

Also a few pep8 fixes in the modified file.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#37316
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

src/sage/rings/ring_extension.pxd

@@ -1,8 +1,8 @@
 from sage.categories.map cimport Map
-from sage.rings.ring cimport CommutativeRing, CommutativeAlgebra
+from sage.rings.ring cimport CommutativeRing
 
 
-cdef class RingExtension_generic(CommutativeAlgebra):
+cdef class RingExtension_generic(CommutativeRing):
     cdef _type
     cdef _backend
     cdef _defining_morphism

src/sage/rings/ring_extension.pyx

@@ -122,7 +122,7 @@ from sage.structure.category_object import normalize_names
 from sage.categories.map cimport Map
 from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.fields import Fields
-from sage.rings.ring cimport CommutativeRing, CommutativeAlgebra
+from sage.rings.ring cimport CommutativeRing
 from sage.rings.integer_ring import ZZ
 from sage.rings.infinity import Infinity
 
@@ -174,8 +174,8 @@ def tower_bases(ring, degree):
           5-adic Field with capped relative precision 20],
          [1, 3, 6])
     """
-    bases = [ ]
-    degrees = [ ]
+    bases = []
+    degrees = []
     base = ring
     deg = 1
     while True:
@@ -454,7 +454,7 @@ class RingExtensionFactory(UniqueFactory):
 
         # We figure out what are the best constructors
         if constructors is None:
-            constructors = [ ]
+            constructors = []
             if gens is not None and len(gens) == 1:
                 constructors.append((RingExtensionWithGen,
                                      {'gen': gens[0], 'names': names,
@@ -498,7 +498,7 @@ RingExtension = RingExtensionFactory("sage.rings.ring_extension.RingExtension")
 # General extensions
 ####################
 
-cdef class RingExtension_generic(CommutativeAlgebra):
+cdef class RingExtension_generic(CommutativeRing):
     r"""
     A generic class for all ring extensions.
 
@@ -580,7 +580,7 @@ cdef class RingExtension_generic(CommutativeAlgebra):
             # but CommutativeRings() seems safer, especially when dealing with
             # morphisms which do not need to preserve the base
             category = CommutativeRings()
-        CommutativeAlgebra.__init__(self, ZZ, category=category)
+        CommutativeRing.__init__(self, ZZ, category=category)
         self._base = base
         self._backend = ring
         self._backend_defining_morphism = defining_morphism
@@ -1135,7 +1135,7 @@ cdef class RingExtension_generic(CommutativeAlgebra):
 
             :meth:`base`, :meth:`absolute_base`, :meth:`is_defined_over`
         """
-        L = [ self ]
+        L = [self]
         base = self
         while isinstance(base, RingExtension_generic):
             base = base.base_ring()
@@ -1358,7 +1358,8 @@ cdef class RingExtension_generic(CommutativeAlgebra):
             (y, x)
         """
         self._check_base(base)
-        return tuple([ self(x) for x in generators(self._backend, backend_parent(self._base)) ])
+        return tuple([self(x) for x in generators(self._backend,
+                                                  backend_parent(self._base))])
 
     def ngens(self, base=None):
         r"""
@@ -2026,7 +2027,6 @@ cdef class RingExtensionFractionField(RingExtension_generic):
              Maximal Order generated by a in Number Field in a
              with defining polynomial x^2 - 2 over its base
             sage: TestSuite(K).run()
-
         """
         RingExtension_generic.__init__(self, defining_morphism, **kwargs)
         if ring is None:
@@ -2138,9 +2138,9 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
             sage: TestSuite(E).run()                                                    # needs sage.rings.number_field
         """
         RingExtension_generic.__init__(self, defining_morphism, **kwargs)
-        self._basis = [ self(b) for b in basis ]
+        self._basis = [self(b) for b in basis]
         if names is None:
-            names = [ ]
+            names = []
             for b in self._basis:
                 b = b._backend
                 if b == 1:
@@ -2154,7 +2154,7 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
             if len(names) != len(self._basis):
                 raise ValueError("the number of names does not match the cardinality of the basis")
         self._basis_names = names
-        self._basis_latex_names = [ latex_variable_name(name) for name in names ]
+        self._basis_latex_names = [latex_variable_name(name) for name in names]
         self._names = tuple(names)
         if check:
             try:
@@ -2335,12 +2335,11 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
             [1, u, u^2]
         """
         if base is self:
-            return [ self.one() ]
-        elif base is self._base:
+            return [self.one()]
+        if base is self._base:
             return self._basis[:]
-        else:
-            b = self._base._basis_over(base)
-            return [ x*y for x in self._basis for y in b ]
+        b = self._base._basis_over(base)
+        return [x * y for x in self._basis for y in b]
 
     def free_module(self, base=None, map=True):
         r"""
@@ -2461,8 +2460,8 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
             sage: L = GF(7^15).over(K)                                                  # needs sage.rings.finite_rings
             sage: for base in L.bases():                                                # needs sage.rings.finite_rings
             ....:     V, i, j = L.free_module(base)
-            ....:     assert([ i(v) for v in V.basis() ] == L.basis_over(base))
-            ....:     assert([ j(x) for x in L.basis_over(base) ] == V.basis())
+            ....:     assert([i(v) for v in V.basis()] == L.basis_over(base))
+            ....:     assert([j(x) for x in L.basis_over(base)] == V.basis())
 
         """
         d = self._degree_over(base)
@@ -2599,15 +2598,15 @@ cdef class RingExtensionWithGen(RingExtensionWithBasis):
         backend_base = backend_parent(defining_morphism.domain())
         _, deg_domain, deg_codomain = common_base(backend_base, defining_morphism.codomain(), True)
         degree = deg_codomain // deg_domain
-        basis_names = [ "" ]
-        basis_latex_names = [ "" ]
+        basis_names = [""]
+        basis_latex_names = [""]
         if degree == 1:
             self._name = None
         else:
-            basis_names += [ self._name ] + [ "%s^%s" % (self._name, i) for i in range(2,degree) ]
+            basis_names += [self._name] + ["%s^%s" % (self._name, i) for i in range(2, degree)]
             latex_name = latex_variable_name(self._name)
-            basis_latex_names += [ latex_name ] + [ "%s^{%s}" % (latex_name, i) for i in range(2,degree) ]
-        basis = [ gen ** i for i in range(degree) ]
+            basis_latex_names += [latex_name] + ["%s^{%s}" % (latex_name, i) for i in range(2, degree)]
+        basis = [gen ** i for i in range(degree)]
         RingExtensionWithBasis.__init__(self, defining_morphism, basis, basis_names, check, **kwargs)
         self._gen = self._backend(gen)
         self._names = (self._name,)
@@ -2681,7 +2680,7 @@ cdef class RingExtensionWithGen(RingExtensionWithBasis):
         from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
         _, _, j = self.free_module(map=True)
         d = self.relative_degree()
-        coeffs = [ -c for c in j(self._gen**d) ] + [ 1 ]
+        coeffs = [-c for c in j(self._gen**d)] + [1]
         S = PolynomialRing(self._base, name=var)
         return S(coeffs)