Adding the category of commutative rings to the category of tensor pr…

…oducts of commutative algebras.

src/sage/categories/commutative_algebras.py

@@ -10,8 +10,11 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.misc.cachefunc import cached_method
 from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
 from sage.categories.algebras import Algebras
+from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.tensor import TensorProductsCategory
 
 class CommutativeAlgebras(CategoryWithAxiom_over_base_ring):
     """
@@ -36,7 +39,7 @@ class CommutativeAlgebras(CategoryWithAxiom_over_base_ring):
         True
         sage: TestSuite(CommutativeAlgebras(ZZ)).run()
 
-    Todo:
+    .. TODO::
 
      - product   ( = Cartesian product)
      - coproduct ( = tensor product over base ring)
@@ -58,3 +61,32 @@ def __contains__(self, A):
         """
         return super().__contains__(A) or \
             (A in Algebras(self.base_ring()) and hasattr(A, "is_commutative") and A.is_commutative())
+
+    class TensorProducts(TensorProductsCategory):
+        """
+        The category of commutative algebras constructed by tensor product of commutative algebras.
+        """
+
+        @cached_method
+        def extra_super_categories(self):
+            """
+            EXAMPLES::
+
+                sage: Algebras(QQ).Commutative().TensorProducts().extra_super_categories()
+                [Category of commutative rings]
+                sage: Algebras(QQ).Commutative().TensorProducts().super_categories()
+                [Category of tensor products of algebras over Rational Field,
+                 Category of commutative algebras over Rational Field]
+
+            TESTS::
+
+                sage: X = algebras.Shuffle(QQ, 'ab')
+                sage: Y = algebras.Shuffle(QQ, 'bc')
+                sage: X in Algebras(QQ).Commutative()
+                True
+                sage: T = tensor([X, Y])
+                sage: T in CommutativeRings()
+                True
+            """
+            return [CommutativeRings()]
+