Merge branch 'u/caruso/21413/class_ring_extension' of git://trac.sage…

…math.org/sage into 21413_ring_extensions

src/sage/categories/action.pxd

@@ -26,3 +26,8 @@ cdef class PrecomposedAction(Action):
 cdef class ActionEndomorphism(Morphism):
     cdef Action _action
     cdef Element _g
+
+cdef class BaseActionOnRing(Action):
+    cdef _base
+    cdef _algebra
+    cdef _defining_morphism

src/sage/categories/action.pyx

@@ -573,3 +573,67 @@ cdef class ActionEndomorphism(Morphism):
                 return (~self._action)(self._g)
 
 
+cdef class BaseActionOnRing(Action):
+    r"""
+    This class implements the left action of the base on the
+    ring given an extension ``ring``/``base``
+
+    See also :func:`RingExtension`
+
+    EXAMPLES::
+
+        sage: K = GF(5^2); z2 = K.gen()
+        sage: L = GF(5^4); z4 = L.gen()
+        sage: E = RingExtension(L, K, K.frobenius_endomorphism())
+
+        sage: z2 * E(z4)   # indirect doctest
+        4*z4^3 + 3*z4^2 + 2*z4 + 2
+        sage: z2^5 * z4
+        4*z4^3 + 3*z4^2 + 2*z4 + 2
+
+    We underline that the action in only defined on the left::
+
+        sage: E(z4) * z2
+        z4^3 + 2*z4^2 + 4*z4 + 3
+        sage: z4 * z2 
+        z4^3 + 2*z4^2 + 4*z4 + 3
+
+    AUTHOR:
+
+    - Xavier Caruso (2016)
+    """
+    def __init__(self, algebra):
+        r"""
+        TESTS::
+
+            sage: from sage.categories.action import BaseActionOnRing
+            sage: BaseActionOnRing(ZZ)
+            Traceback (most recent call last):
+            ...
+            TypeError: Integer Ring is not an instance of AlgebraFromMorphism
+        """
+        from sage.rings.algebra_from_morphism import AlgebraFromMorphism
+        if not isinstance(algebra, AlgebraFromMorphism):
+            raise TypeError("%s is not an instance of AlgebraFromMorphism" % algebra)
+        self._base = algebra.base_ring()
+        self._algebra = algebra
+        self._defining_morphism = algebra.defining_morphism()
+
+        import operator
+        Action.__init__(self, self._base, algebra, op=operator.mul)
+
+    cpdef _call_(self, scalar, element):
+        r"""
+        Return the action of ``scalar`` on ``element``
+
+        EXAMPLE::
+
+            sage: K = GF(5^2); z2 = K.gen()
+            sage: L = GF(5^4); z4 = L.gen()
+            sage: E = RingExtension(L, K, K.frobenius_endomorphism())
+
+            sage: z2 * E(z4)   # indirect doctest
+            4*z4^3 + 3*z4^2 + 2*z4 + 2
+        """
+        result = self._defining_morphism(scalar) * element.element_in_ring()
+        return self._algebra(result)