gh-37163: moving modular stuff to "CommutativeRing" class
    
as a step towards simplifying old coercion framework

also adding some typing annotations

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: #37163
Reported by: Frédéric Chapoton
Reviewer(s): Alex J Best

Author: Release Manager

src/sage/modular/hecke/algebra.py

@@ -26,14 +26,16 @@
 #
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
+from typing import Iterator
 
-import sage.rings.infinity
+from sage.rings.infinity import infinity
+from sage.categories.algebras import Algebras
 from sage.matrix.constructor import matrix
 from sage.arith.functions import lcm
 from sage.arith.misc import gcd
 from sage.misc.latex import latex
 from sage.matrix.matrix_space import MatrixSpace
-from sage.rings.ring import CommutativeAlgebra
+from sage.rings.ring import CommutativeRing
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
 from sage.structure.element import Element
@@ -42,7 +44,7 @@
 from sage.structure.richcmp import richcmp_method, richcmp
 
 
-def is_HeckeAlgebra(x):
+def is_HeckeAlgebra(x) -> bool:
     r"""
     Return ``True`` if x is of type HeckeAlgebra.
 
@@ -105,7 +107,7 @@ def _heckebasis(M):
 
 
 @richcmp_method
-class HeckeAlgebra_base(CachedRepresentation, CommutativeAlgebra):
+class HeckeAlgebra_base(CachedRepresentation, CommutativeRing):
     """
     Base class for algebras of Hecke operators on a fixed Hecke module.
 
@@ -163,7 +165,7 @@ def __classcall__(cls, M):
             pass
         return super().__classcall__(cls, M)
 
-    def __init__(self, M):
+    def __init__(self, M) -> None:
         """
         Initialization.
 
@@ -179,7 +181,9 @@ def __init__(self, M):
         if not module.is_HeckeModule(M):
             raise TypeError("M (=%s) must be a HeckeModule" % M)
         self.__M = M
-        CommutativeAlgebra.__init__(self, M.base_ring())
+        cat = Algebras(M.base_ring()).Commutative()
+        CommutativeRing.__init__(self, base_ring=M.base_ring(),
+                                 category=cat)
 
     def _an_element_(self):
         r"""
@@ -319,12 +323,13 @@ def ngens(self):
             sage: CuspForms(1, 12).anemic_hecke_algebra().ngens()
             +Infinity
         """
-        return sage.rings.infinity.infinity
+        return infinity
 
-    def is_noetherian(self):
+    def is_noetherian(self) -> bool:
         """
-        Return ``True`` if this Hecke algebra is Noetherian as a ring. This is true
-        if and only if the base ring is Noetherian.
+        Return ``True`` if this Hecke algebra is Noetherian as a ring.
+
+        This is true if and only if the base ring is Noetherian.
 
         EXAMPLES::
 
@@ -343,9 +348,9 @@ def matrix_space(self):
             sage: CuspForms(3, 24, base_ring=Qp(5)).anemic_hecke_algebra().matrix_space()
             Full MatrixSpace of 7 by 7 dense matrices over 5-adic Field with capped relative precision 20
         """
-        return sage.matrix.matrix_space.MatrixSpace(self.base_ring(), self.module().rank())
+        return MatrixSpace(self.base_ring(), self.module().rank())
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         LaTeX representation of self.
 
@@ -569,7 +574,7 @@ class HeckeAlgebra_full(HeckeAlgebra_base):
     A full Hecke algebra (including the operators `T_n` where `n` is not
     assumed to be coprime to the level).
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         String representation of self.
 
@@ -580,7 +585,7 @@ def _repr_(self):
         """
         return "Full Hecke algebra acting on %s" % self.module()
 
-    def __richcmp__(self, other, op):
+    def __richcmp__(self, other, op) -> bool:
         r"""
         Compare self to other.
 
@@ -631,15 +636,15 @@ class HeckeAlgebra_anemic(HeckeAlgebra_base):
     r"""
     An anemic Hecke algebra, generated by Hecke operators with index coprime to the level.
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         EXAMPLES::
 
             sage: H = CuspForms(3, 12).anemic_hecke_algebra()._repr_()
         """
         return "Anemic Hecke algebra acting on %s" % self.module()
 
-    def __richcmp__(self, other, op):
+    def __richcmp__(self, other, op) -> bool:
         r"""
         Compare self to other.
 
@@ -689,7 +694,7 @@ def is_anemic(self) -> bool:
         """
         return True
 
-    def gens(self):
+    def gens(self) -> Iterator:
         r"""
         Return a generator over all Hecke operator `T_n` for
         `n = 1, 2, 3, \ldots`, with `n` coprime to the