gh-39303: let the categories handle "is_commutative" for rings
    
This makes sure that the method `is_commutative` for rings is handled by
the category framework and not by the auld `Ring` class.

Also adding annotations on some `is_commutative` methods

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.
    
URL: #39303
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -130,7 +130,6 @@ This base class provides a lot more methods than a general parent::
      'gen',
      'gens',
      'integral_closure',
-     'is_commutative',
      'is_field',
      'krull_dimension',
      'ngens',

src/sage/algebras/clifford_algebra.py

@@ -844,7 +844,7 @@ def one_basis(self):
         """
         return FrozenBitset()
 
-    def is_commutative(self):
+    def is_commutative(self) -> bool:
         """
         Check if ``self`` is a commutative algebra.
 

src/sage/algebras/finite_dimensional_algebras/finite_dimensional_algebra.py

@@ -497,7 +497,7 @@ def is_associative(self):
         return True
 
     @cached_method
-    def is_commutative(self):
+    def is_commutative(self) -> bool:
         """
         Return ``True`` if ``self`` is commutative.
 

src/sage/algebras/iwahori_hecke_algebra.py

@@ -766,7 +766,7 @@ def is_field(self, proof=True):
                 """
                 return False
 
-            def is_commutative(self):
+            def is_commutative(self) -> bool:
                 """
                 Return whether this Iwahori-Hecke algebra is commutative.
 

src/sage/algebras/steenrod/steenrod_algebra.py

@@ -2827,10 +2827,12 @@ def gen(self, i=0):
             tot += 1
         return test
 
-    def is_commutative(self):
+    def is_commutative(self) -> bool:
         r"""
         Return ``True`` if ``self`` is graded commutative, as determined by the
-        profile function.  In particular, a sub-Hopf algebra of the
+        profile function.
+
+        In particular, a sub-Hopf algebra of the
         mod 2 Steenrod algebra is commutative if and only if there is
         an integer `n>0` so that its profile function `e` satisfies
 

src/sage/categories/commutative_rings.py

@@ -46,6 +46,17 @@ class CommutativeRings(CategoryWithAxiom):
 
         sage: GroupAlgebra(CyclicPermutationGroup(3), QQ) in CommutativeRings()     # not implemented, needs sage.groups sage.modules
         True
+
+    Some tests for the method ``is_commutative``::
+
+        sage: QQ.is_commutative()
+        True
+        sage: ZpCA(7).is_commutative()                                              # needs sage.rings.padics
+        True
+        sage: A = QuaternionAlgebra(QQ, -1, -3, names=('i','j','k')); A             # needs sage.combinat sage.modules
+        Quaternion Algebra (-1, -3) with base ring Rational Field
+        sage: A.is_commutative()                                                    # needs sage.combinat sage.modules
+        False
     """
     class ParentMethods:
         def is_commutative(self) -> bool:

src/sage/categories/finite_dimensional_algebras_with_basis.py

@@ -1143,7 +1143,7 @@ def is_identity_decomposition_into_orthogonal_idempotents(self, l):
                                                   for f in l[:i]))
 
         @cached_method
-        def is_commutative(self):
+        def is_commutative(self) -> bool:
             """
             Return whether ``self`` is a commutative algebra.
 
@@ -1158,7 +1158,7 @@ def is_commutative(self):
                 True
             """
             B = list(self.basis())
-            try: # See if 1 is a basis element, if so, remove it
+            try:  # See if 1 is a basis element, if so, remove it
                 B.remove(self.one())
             except ValueError:
                 pass

src/sage/categories/lie_algebras.py

@@ -589,10 +589,11 @@ def is_abelian(self):
             zero = self.zero()
             return all(x._bracket_(y) == zero for x in G for y in G)
 
-        def is_commutative(self):
+        def is_commutative(self) -> bool:
             """
-            Return if ``self`` is commutative. This is equivalent to ``self``
-            being abelian.
+            Return if ``self`` is commutative.
+
+            This is equivalent to ``self`` being abelian.
 
             EXAMPLES::
 

src/sage/categories/magmas.py

@@ -398,7 +398,7 @@ def is_field(self, proof=True):
     class Commutative(CategoryWithAxiom):
 
         class ParentMethods:
-            def is_commutative(self):
+            def is_commutative(self) -> bool:
                 """
                 Return ``True``, since commutative magmas are commutative.
 

src/sage/rings/quotient_ring.py

@@ -589,7 +589,7 @@ def _latex_(self):
         """
         return "%s/%s" % (latex.latex(self.cover_ring()), latex.latex(self.defining_ideal()))
 
-    def is_commutative(self):
+    def is_commutative(self) -> bool:
         """
         Tell whether this quotient ring is commutative.