gh-37158: use Parent in quotient rings too
    
instead of using the deprecated `ParentWithGens`

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: #37158
Reported by: Frédéric Chapoton
Reviewer(s):

Author: Release Manager

src/sage/categories/commutative_rings.py

@@ -140,12 +140,13 @@ def is_commutative(self) -> bool:
 
         def _ideal_class_(self, n=0):
             r"""
-            Return a callable object that can be used to create ideals in this
-            commutative ring.
+            Return a callable object that can be used to create ideals
+            in this commutative ring.
 
-            This class can depend on `n`, the number of generators of the ideal.
-            The default input of `n=0` indicates an unspecified number of generators,
-            in which case a class that works for any number of generators is returned.
+            This class can depend on `n`, the number of generators of
+            the ideal.  The default input of `n=0` indicates an
+            unspecified number of generators, in which case a class
+            that works for any number of generators is returned.
 
             EXAMPLES::
 

src/sage/categories/finite_dimensional_algebras_with_basis.py

@@ -504,6 +504,12 @@ def principal_ideal(self, a, side='left', *args, **opts):
             r"""
             Construct the ``side`` principal ideal generated by ``a``.
 
+            INPUT:
+
+            - ``a`` -- an element
+            - ``side`` -- ``left`` (default) or ``right``
+            - ``coerce`` -- ignored, for compatibility with categories
+
             EXAMPLES:
 
             In order to highlight the difference between left and
@@ -547,6 +553,7 @@ def principal_ideal(self, a, side='left', *args, **opts):
 
                 - :meth:`peirce_summand`
             """
+            opts.pop("coerce", None)
             return self.submodule([(a * b if side == 'right' else b * a)
                                    for b in self.basis()], *args, **opts)
 

src/sage/categories/rings.py

@@ -5,7 +5,8 @@
 # ****************************************************************************
 #  Copyright (C) 2005      David Kohel <[email protected]>
 #                          William Stein <[email protected]>
-#                2008      Teresa Gomez-Diaz (CNRS) <[email protected]>
+#                2008      Teresa Gomez-Diaz (CNRS)
+#                          <[email protected]>
 #                2008-2011 Nicolas M. Thiery <nthiery at users.sf.net>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
@@ -751,109 +752,6 @@ def __pow__(self, n):
                 from sage.modules.free_module import FreeModule
                 return FreeModule(self, n)
 
-        @cached_method
-        def ideal_monoid(self):
-            """
-            The monoid of the ideals of this ring.
-
-            EXAMPLES::
-
-                sage: # needs sage.modules
-                sage: MS = MatrixSpace(QQ, 2, 2)
-                sage: isinstance(MS, Ring)
-                False
-                sage: MS in Rings()
-                True
-                sage: MS.ideal_monoid()
-                Monoid of ideals of Full MatrixSpace of 2 by 2 dense matrices
-                over Rational Field
-
-            Note that the monoid is cached::
-
-                sage: MS.ideal_monoid() is MS.ideal_monoid()                            # needs sage.modules
-                True
-
-            More examples::
-
-                sage: # needs sage.combinat sage.modules
-                sage: F.<x,y,z> = FreeAlgebra(ZZ, 3)
-                sage: I = F * [x*y + y*z, x^2 + x*y - y*x - y^2] * F
-                sage: Q = F.quotient(I)
-                sage: Q.ideal_monoid()
-                Monoid of ideals of Quotient of Free Algebra on 3 generators (x, y, z)
-                 over Integer Ring by the ideal (x*y + y*z, x^2 + x*y - y*x - y^2)
-                sage: F.<x,y,z> = FreeAlgebra(ZZ, implementation='letterplace')
-                sage: I = F * [x*y + y*z, x^2 + x*y - y*x - y^2] * F
-                sage: Q = F.quo(I)
-                sage: Q.ideal_monoid()
-                Monoid of ideals of Quotient of Free Associative Unital Algebra
-                 on 3 generators (x, y, z) over Integer Ring
-                 by the ideal (x*y + y*z, x*x + x*y - y*x - y*y)
-
-                sage: ZZ.ideal_monoid()
-                Monoid of ideals of Integer Ring
-                sage: R.<x> = QQ[]; R.ideal_monoid()
-                Monoid of ideals of Univariate Polynomial Ring in x over Rational Field
-            """
-            try:
-                from sage.rings.ideal_monoid import IdealMonoid
-                return IdealMonoid(self)
-            except TypeError:
-                from sage.rings.noncommutative_ideals import IdealMonoid_nc
-                return IdealMonoid_nc(self)
-
-        def _ideal_class_(self, n=0):
-            r"""
-            Return a callable object that can be used to create ideals in this
-            ring.
-
-            EXAMPLES::
-
-                sage: MS = MatrixSpace(QQ, 2, 2)                                        # needs sage.modules
-                sage: MS._ideal_class_()                                                # needs sage.modules
-                <class 'sage.rings.noncommutative_ideals.Ideal_nc'>
-
-            Since :issue:`7797`, non-commutative rings have ideals as well::
-
-                sage: A = SteenrodAlgebra(2)                                                # needs sage.combinat sage.modules
-                sage: A._ideal_class_()                                                     # needs sage.combinat sage.modules
-                <class 'sage.rings.noncommutative_ideals.Ideal_nc'>
-            """
-            from sage.rings.noncommutative_ideals import Ideal_nc
-            return Ideal_nc
-
-        @cached_method
-        def zero_ideal(self):
-            """
-            Return the zero ideal of this ring (cached).
-
-            EXAMPLES::
-
-                sage: ZZ.zero_ideal()
-                Principal ideal (0) of Integer Ring
-                sage: QQ.zero_ideal()
-                Principal ideal (0) of Rational Field
-                sage: QQ['x'].zero_ideal()
-                Principal ideal (0) of Univariate Polynomial Ring in x over Rational Field
-
-            The result is cached::
-
-                sage: ZZ.zero_ideal() is ZZ.zero_ideal()
-                True
-
-            TESTS:
-
-            Make sure that :issue:`13644` is fixed::
-
-                sage: # needs sage.rings.padics
-                sage: K = Qp(3)
-                sage: R.<a> = K[]
-                sage: L.<a> = K.extension(a^2-3)
-                sage: L.ideal(a)
-                Principal ideal (1 + O(a^40)) of 3-adic Eisenstein Extension Field in a defined by a^2 - 3
-            """
-            return self._ideal_class_(1)(self, [self.zero()])
-
         @cached_method
         def unit_ideal(self):
             """
@@ -866,21 +764,6 @@ def unit_ideal(self):
             """
             return self._ideal_class_(1)(self, [self.one()])
 
-        def principal_ideal(self, gen, coerce=True):
-            """
-            Return the principal ideal generated by gen.
-
-            EXAMPLES::
-
-                sage: R.<x,y> = ZZ[]
-                sage: R.principal_ideal(x+2*y)
-                Ideal (x + 2*y) of Multivariate Polynomial Ring in x, y over Integer Ring
-            """
-            C = self._ideal_class_(1)
-            if coerce:
-                gen = self(gen)
-            return C(self, [gen])
-
         def characteristic(self):
             """
             Return the characteristic of this ring.

src/sage/categories/rngs.py

@@ -2,14 +2,15 @@
 r"""
 Rngs
 """
-#*****************************************************************************
+# ****************************************************************************
 #  Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <[email protected]>
 #                2012 Nicolas M. Thiery <nthiery at users.sf.net>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
-#                  http://www.gnu.org/licenses/
-#******************************************************************************
+#                  https://www.gnu.org/licenses/
+# *****************************************************************************
 
+from sage.misc.cachefunc import cached_method
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.misc.lazy_import import LazyImport
 from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas
@@ -49,3 +50,111 @@ class Rngs(CategoryWithAxiom):
     _base_category_class_and_axiom = (MagmasAndAdditiveMagmas.Distributive.AdditiveAssociative.AdditiveCommutative.AdditiveUnital.Associative, "AdditiveInverse")
 
     Unital = LazyImport('sage.categories.rings', 'Rings', at_startup=True)
+
+    class ParentMethods:
+
+        @cached_method
+        def ideal_monoid(self):
+            """
+            The monoid of the ideals of this ring.
+
+            .. NOTE::
+
+                The code is copied from the base class of rings.
+                This is since there are rings that do not inherit
+                from that class, such as matrix algebras.  See
+                :issue:`7797`.
+
+            EXAMPLES::
+
+                sage: # needs sage.modules
+                sage: MS = MatrixSpace(QQ, 2, 2)
+                sage: isinstance(MS, Ring)
+                False
+                sage: MS in Rings()
+                True
+                sage: MS.ideal_monoid()
+                Monoid of ideals of Full MatrixSpace of 2 by 2 dense matrices
+                over Rational Field
+
+            Note that the monoid is cached::
+
+                sage: MS.ideal_monoid() is MS.ideal_monoid()                            # needs sage.modules
+                True
+            """
+            try:
+                from sage.rings.ideal_monoid import IdealMonoid
+                return IdealMonoid(self)
+            except TypeError:
+                from sage.rings.noncommutative_ideals import IdealMonoid_nc
+                return IdealMonoid_nc(self)
+
+        def _ideal_class_(self, n=0):
+            r"""
+            Return a callable object that can be used to create ideals in this
+            ring.
+
+            The argument `n`, standing for the number of generators
+            of the ideal, is ignored.
+
+            EXAMPLES::
+
+                sage: MS = MatrixSpace(QQ, 2, 2)                                        # needs sage.modules
+                sage: MS._ideal_class_()                                                # needs sage.modules
+                <class 'sage.rings.noncommutative_ideals.Ideal_nc'>
+
+            Since :issue:`7797`, non-commutative rings have ideals as well::
+
+                sage: A = SteenrodAlgebra(2)                                                # needs sage.combinat sage.modules
+                sage: A._ideal_class_()                                                     # needs sage.combinat sage.modules
+                <class 'sage.rings.noncommutative_ideals.Ideal_nc'>
+            """
+            from sage.rings.noncommutative_ideals import Ideal_nc
+            return Ideal_nc
+
+        def principal_ideal(self, gen, coerce=True):
+            """
+            Return the principal ideal generated by ``gen``.
+
+            EXAMPLES::
+
+                sage: R.<x,y> = ZZ[]
+                sage: R.principal_ideal(x+2*y)
+                Ideal (x + 2*y) of Multivariate Polynomial Ring in x, y over Integer Ring
+            """
+            C = self._ideal_class_(1)
+            if coerce:
+                gen = self(gen)
+            return C(self, [gen])
+
+        @cached_method
+        def zero_ideal(self):
+            """
+            Return the zero ideal of this ring (cached).
+
+            EXAMPLES::
+
+                sage: ZZ.zero_ideal()
+                Principal ideal (0) of Integer Ring
+                sage: QQ.zero_ideal()
+                Principal ideal (0) of Rational Field
+                sage: QQ['x'].zero_ideal()
+                Principal ideal (0) of Univariate Polynomial Ring in x over Rational Field
+
+            The result is cached::
+
+                sage: ZZ.zero_ideal() is ZZ.zero_ideal()
+                True
+
+            TESTS:
+
+            Make sure that :issue:`13644` is fixed::
+
+                sage: # needs sage.rings.padics
+                sage: K = Qp(3)
+                sage: R.<a> = K[]
+                sage: L.<a> = K.extension(a^2-3)
+                sage: L.ideal(a)
+                Principal ideal (1 + O(a^40)) of 3-adic Eisenstein Extension Field in a defined by a^2 - 3
+            """
+            return self._ideal_class_(1)(self, [self.zero()])

src/sage/rings/finite_rings/finite_field_prime_modn.py

@@ -300,6 +300,30 @@ def gen(self, n=0):
             self.__gen = self.one()
         return self.__gen
 
+    def gens(self) -> tuple:
+        r"""
+        Return a tuple containing the generator of ``self``.
+
+        .. WARNING::
+
+            The generator is not guaranteed to be a generator for the
+            multiplicative group.  To obtain the latter, use
+            :meth:`~sage.rings.finite_rings.finite_field_base.FiniteFields.multiplicative_generator()`
+            or use the ``modulus="primitive"`` option when constructing
+            the field.
+
+        EXAMPLES::
+
+            sage: k = GF(1009, modulus='primitive')
+            sage: k.gens()
+            (11,)
+
+            sage: k = GF(1009)
+            sage: k.gens()
+            (1,)
+        """
+        return (self.gen(),)
+
     def __iter__(self):
         """
         Return an iterator over ``self``.

src/sage/rings/function_field/ideal.py

@@ -1066,7 +1066,7 @@ def __init__(self, R):
             sage: M = O.ideal_monoid()
             sage: TestSuite(M).run()
         """
-        self.Element = R._ideal_class
+        self.Element = R._ideal_class_
         Parent.__init__(self, category=Monoids())
 
         self.__R = R

src/sage/rings/function_field/order.py

@@ -145,7 +145,7 @@ def __init__(self, field, ideal_class=FunctionFieldIdeal, category=None):
         category = IntegralDomains().or_subcategory(category).Infinite()
         Parent.__init__(self, category=category, facade=field)
 
-        self._ideal_class = ideal_class # element class for parent ideal monoid
+        self._ideal_class_ = ideal_class  # element class for parent ideal monoid
         self._field = field
 
     def is_field(self, proof=True):

src/sage/rings/function_field/order_polymod.py

@@ -649,9 +649,7 @@ def decomposition(self, ideal):
         # algebra O mod pO.
         matrices_reduced = [M.mod(p) for M in matrices]
         cat = CommutativeAlgebras(k).FiniteDimensional().WithBasis()
-        A = FiniteDimensionalAlgebra(k, matrices_reduced,
-                                     assume_associative=True,
-                                     category=cat)
+        A = FiniteDimensionalAlgebra(k, matrices_reduced, category=cat)
 
         # Each prime ideal of the algebra A corresponds to a prime ideal of O,
         # and since the algebra is an Artinian ring, all of its prime ideals

src/sage/rings/morphism.pyx

@@ -409,6 +409,7 @@ from sage.rings import ideal
 import sage.structure.all
 from sage.structure.richcmp cimport (richcmp, rich_to_bool)
 from sage.misc.cachefunc import cached_method
+from sage.categories.rings import Rings
 from sage.categories.facade_sets import FacadeSets
 
 
@@ -1288,7 +1289,8 @@ cdef class RingHomomorphism(RingMap):
         from sage.rings.ideal import Ideal_generic
         A = self.domain()
         B = self.codomain()
-        if not (A.is_commutative() and B.is_commutative()):
+        Comm = Rings().Commutative()
+        if not (A in Comm and B in Comm):
             raise NotImplementedError("rings are not commutative")
         if A.base_ring() != B.base_ring():
             raise NotImplementedError("base rings must be equal")