sagemathgh-37719: Refactor ring categories

    
A somewhat large refactoring of some of the auld classes for rings, and
related categories

- introducing a new category of Noetherian rings
- moving some methods in appropriate categories
- fixing necessary details

### 📝 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 accordingly.

Depends on sagemath#37778
    
URL: sagemath#37719
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe, Travis Scrimshaw

pkgs/sagemath-categories/known-test-failures.json

@@ -597,6 +597,10 @@
         "failed": true,
         "ntests": 126
     },
+    "sage.categories.noetherian_rings": {
+        "failed": true,
+        "ntests": 19
+    },
     "sage.categories.number_fields": {
         "failed": true,
         "ntests": 41
@@ -1428,4 +1432,4 @@
     "sage.typeset.unicode_characters": {
         "ntests": 27
     }
-}
+}

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -116,7 +116,6 @@ This base class provides a lot more methods than a general parent::
      '_coerce_impl',
      '_default_category',
      '_gens',
-     '_ideal_class_',
      '_ideal_monoid',
      '_latex_names',
      '_list',
@@ -127,24 +126,20 @@ This base class provides a lot more methods than a general parent::
      '_zero_ideal',
      'algebraic_closure',
      'base_extend',
-     'class_group',
-     'content',
      'derivation',
      'derivation_module',
      'divides',
      'epsilon',
      'extension',
      'fraction_field',
      'frobenius_endomorphism',
-     'gcd',
      'gen',
      'gens',
      'ideal',
      'ideal_monoid',
      'integral_closure',
      'is_commutative',
      'is_field',
-     'is_integral_domain',
      'is_integrally_closed',
      'is_noetherian',
      'is_prime_field',
@@ -858,7 +853,9 @@ The four axioms requested for coercions
       rational field is a homomorphism of euclidean domains::
 
           sage: QQ.coerce_map_from(ZZ).category_for()
-          Join of Category of euclidean domains and Category of infinite sets
+          Join of Category of euclidean domains
+          and Category of noetherian rings
+          and Category of infinite sets
           and Category of metric spaces
 
       .. end of output

src/doc/en/tutorial/tour_coercion.rst

@@ -118,6 +118,7 @@ implemented in Sage as well:
     sage: ZZ.category()
     Join of Category of Dedekind domains
         and Category of euclidean domains
+        and Category of noetherian rings
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/doc/fr/tutorial/tour_coercion.rst

@@ -119,6 +119,7 @@ par ailleurs les catégories en tant que telles :
     sage: ZZ.category()
     Join of Category of Dedekind domains
         and Category of euclidean domains
+        and Category of noetherian rings
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/doc/ja/tutorial/tour_coercion.rst

@@ -101,6 +101,7 @@ Sageのクラス階層と圏の階層構造にはそれなりに類似が見ら
     sage: ZZ.category()
     Join of Category of Dedekind domains
         and Category of euclidean domains
+        and Category of noetherian rings
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/doc/pt/tutorial/tour_coercion.rst

@@ -125,6 +125,7 @@ categorias matemáticas também são implementadas no Sage:
     sage: ZZ.category()
     Join of Category of Dedekind domains
         and Category of euclidean domains
+        and Category of noetherian rings
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/sage/algebras/clifford_algebra.py

@@ -514,6 +514,7 @@ def __init__(self, Q, names, category=None):
             sage: Cl.category()
             Category of finite dimensional super algebras with basis over
              (Dedekind domains and euclidean domains
+              and noetherian rings
               and infinite enumerated sets and metric spaces)
             sage: TestSuite(Cl).run()
 
@@ -1093,6 +1094,7 @@ def lift_module_morphism(self, m, names=None):
             sage: phi.category_for()
             Category of finite dimensional super algebras with basis over
              (Dedekind domains and euclidean domains
+              and noetherian rings
               and infinite enumerated sets and metric spaces)
             sage: phi.matrix()
             [  1   0   0   0   7  -3  -7   0]
@@ -1177,6 +1179,7 @@ def lift_isometry(self, m, names=None):
             sage: phi.category_for()
             Category of finite dimensional super algebras with basis over
              (Dedekind domains and euclidean domains
+              and noetherian rings
               and infinite enumerated sets and metric spaces)
             sage: phi.matrix()
             [ 1  0  0  0  1  2  5  0]

src/sage/algebras/quatalg/quaternion_algebra.py

@@ -515,7 +515,7 @@ def is_integral_domain(self, proof=True) -> bool:
 
     def is_noetherian(self) -> bool:
         """
-        Return ``True`` always, since any quaternion algebra is a noetherian
+        Return ``True`` always, since any quaternion algebra is a Noetherian
         ring (because it is a finitely generated module over a field).
 
         EXAMPLES::

src/sage/algebras/steenrod/steenrod_algebra.py

@@ -3054,7 +3054,7 @@ def is_integral_domain(self, proof=True):
 
     def is_noetherian(self):
         """
-        This algebra is noetherian if and only if it is finite.
+        This algebra is Noetherian if and only if it is finite.
 
         EXAMPLES::
 

src/sage/categories/bimodules.py

@@ -77,6 +77,7 @@ def _make_named_class_key(self, name):
                  and Category of metric spaces,
              Join of Category of Dedekind domains
                  and Category of euclidean domains
+                 and Category of noetherian rings
                  and Category of infinite enumerated sets
                  and Category of metric spaces)
 
@@ -85,6 +86,7 @@ def _make_named_class_key(self, name):
             (Category of fields,
              Join of Category of Dedekind domains
              and Category of euclidean domains
+             and Category of noetherian rings
              and Category of infinite enumerated sets
              and Category of metric spaces)
 

src/sage/categories/category.py

@@ -1228,7 +1228,7 @@ def structure(self):
             sage: structure(Rings())
             (Category of unital magmas, Category of additive unital additive magmas)
             sage: structure(Fields())
-            (Category of euclidean domains,)
+            (Category of euclidean domains, Category of noetherian rings)
             sage: structure(Algebras(QQ))
             (Category of unital magmas,
              Category of right modules over Rational Field,
@@ -2840,6 +2840,7 @@ def _make_named_class_key(self, name):
             sage: Algebras(ZZ)._make_named_class_key("parent_class")
             Join of Category of Dedekind domains
                  and Category of euclidean domains
+                 and Category of noetherian rings
                  and Category of infinite enumerated sets
                  and Category of metric spaces
 
@@ -2852,6 +2853,7 @@ def _make_named_class_key(self, name):
                  and Category of metric spaces,
              Join of Category of Dedekind domains
                  and Category of euclidean domains
+                 and Category of noetherian rings
                  and Category of infinite enumerated sets
                  and Category of metric spaces)
 
@@ -2979,6 +2981,7 @@ def _make_named_class_key(self, name):
             sage: Modules(ZZ)._make_named_class_key('element_class')
             Join of Category of Dedekind domains
                  and Category of euclidean domains
+                 and Category of noetherian rings
                  and Category of infinite enumerated sets
                  and Category of metric spaces
             sage: Modules(QQ)._make_named_class_key('parent_class')

src/sage/categories/category_types.py

@@ -226,7 +226,8 @@ def _make_named_class_key(self, name):
 
             sage: Modules(ZZ)._make_named_class_key('element_class')
             Join of Category of Dedekind domains
-             and  Category of euclidean domains
+             and Category of euclidean domains
+             and Category of noetherian rings
              and Category of infinite enumerated sets
              and Category of metric spaces
             sage: Modules(QQ)._make_named_class_key('parent_class')

src/sage/categories/commutative_rings.py

@@ -65,6 +65,39 @@ def is_commutative(self) -> bool:
             """
             return True
 
+        def _ideal_class_(self, n=0):
+            r"""
+            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.
+
+            EXAMPLES::
+
+                sage: ZZ._ideal_class_()
+                <class 'sage.rings.ideal.Ideal_pid'>
+                sage: RR._ideal_class_()
+                <class 'sage.rings.ideal.Ideal_pid'>
+                sage: R.<x,y> = GF(5)[]
+                sage: R._ideal_class_(1)
+                <class 'sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal'>
+                sage: S = R.quo(x^3 - y^2)
+                sage: S._ideal_class_(1)
+                <class 'sage.rings.quotient_ring.QuotientRingIdeal_principal'>
+                sage: S._ideal_class_(2)
+                <class 'sage.rings.quotient_ring.QuotientRingIdeal_generic'>
+                sage: T.<z> = S[]                                                           # needs sage.libs.singular
+                sage: T._ideal_class_(5)                                                    # needs sage.libs.singular
+                <class 'sage.rings.ideal.Ideal_generic'>
+                sage: T._ideal_class_(1)                                                    # needs sage.libs.singular
+                <class 'sage.rings.ideal.Ideal_principal'>
+            """
+            # One might need more than just n
+            from sage.rings.ideal import Ideal_generic, Ideal_principal
+            return Ideal_principal if n == 1 else Ideal_generic
+
         def _test_divides(self, **options):
             r"""
             Run generic tests on the method :meth:`divides`.
@@ -248,6 +281,18 @@ class Finite(CategoryWithAxiom):
             ....:                    GF(5)]) in Rings().Commutative().Finite()
             True
         """
+        def extra_super_categories(self):
+            r"""
+            Let Sage know that finite commutative rings are Noetherian.
+
+            EXAMPLES::
+
+                sage: CommutativeRings().Finite().extra_super_categories()
+                [Category of noetherian rings]
+            """
+            from sage.categories.noetherian_rings import NoetherianRings
+            return [NoetherianRings()]
+
         class ParentMethods:
             def cyclotomic_cosets(self, q, cosets=None):
                 r"""
@@ -367,7 +412,7 @@ def cyclotomic_cosets(self, q, cosets=None):
                 try:
                     ~q
                 except ZeroDivisionError:
-                    raise ValueError("%s is not invertible in %s" % (q,self))
+                    raise ValueError("%s is not invertible in %s" % (q, self))
 
                 if cosets is None:
                     rest = set(self)
@@ -378,7 +423,7 @@ def cyclotomic_cosets(self, q, cosets=None):
                 while rest:
                     x0 = rest.pop()
                     o = [x0]
-                    x = q*x0
+                    x = q * x0
                     while x != x0:
                         o.append(x)
                         rest.discard(x)

src/sage/categories/fields.py

@@ -18,6 +18,7 @@
 from sage.categories.category_singleton import Category_contains_method_by_parent_class
 from sage.categories.euclidean_domains import EuclideanDomains
 from sage.categories.division_rings import DivisionRings
+from sage.categories.noetherian_rings import NoetherianRings
 
 from sage.structure.element import coerce_binop
 
@@ -33,7 +34,9 @@ class Fields(CategoryWithAxiom):
         sage: K
         Category of fields
         sage: Fields().super_categories()
-        [Category of euclidean domains, Category of division rings]
+        [Category of euclidean domains,
+         Category of division rings,
+         Category of noetherian rings]
 
         sage: K(IntegerRing())
         Rational Field
@@ -54,10 +57,9 @@ def extra_super_categories(self):
         EXAMPLES::
 
             sage: Fields().extra_super_categories()
-            [Category of euclidean domains]
-
+            [Category of euclidean domains, Category of noetherian rings]
         """
-        return [EuclideanDomains()]
+        return [EuclideanDomains(), NoetherianRings()]
 
     def __contains__(self, x):
         """
@@ -165,7 +167,9 @@ def _call_(self, x):
             sage: K
             Category of fields
             sage: Fields().super_categories()
-            [Category of euclidean domains, Category of division rings]
+            [Category of euclidean domains,
+             Category of division rings,
+             Category of noetherian rings]
 
             sage: K(IntegerRing())  # indirect doctest
             Rational Field

src/sage/categories/group_algebras.py

@@ -358,7 +358,7 @@ def is_integral_domain(self, proof=True):
             return ans
 
         # I haven't written is_noetherian(), because I don't know when group
-        # algebras are noetherian, and I haven't written is_prime_field(), because
+        # algebras are Noetherian, and I haven't written is_prime_field(), because
         # I don't know if that means "is canonically isomorphic to a prime field"
         # or "is identical to a prime field".
 

src/sage/categories/homset.py

@@ -1245,7 +1245,7 @@ def reversed(self):
              the principal ideal domain Integer Ring to Ambient free module
              of rank 3 over the principal ideal domain Integer Ring in
              Category of finite dimensional modules with basis over (Dedekind
-             domains and euclidean domains
+             domains and euclidean domains and noetherian rings
              and infinite enumerated sets and metric spaces)
             sage: type(H)
             <class 'sage.modules.free_module_homspace.FreeModuleHomspace_with_category'>
@@ -1254,7 +1254,7 @@ def reversed(self):
              the principal ideal domain Integer Ring to Ambient free module
              of rank 2 over the principal ideal domain Integer Ring in
              Category of finite dimensional modules with basis over (Dedekind
-             domains and euclidean domains
+             domains and euclidean domains and noetherian rings
              and infinite enumerated sets and metric spaces)
             sage: type(H.reversed())
             <class 'sage.modules.free_module_homspace.FreeModuleHomspace_with_category'>

src/sage/categories/integral_domains.py

@@ -103,6 +103,8 @@ def is_integral_domain(self, proof=True):
 
             EXAMPLES::
 
+                sage: ZZ.is_integral_domain()
+                True
                 sage: QQ.is_integral_domain()
                 True
                 sage: Parent(QQ, category=IntegralDomains()).is_integral_domain()
@@ -113,6 +115,9 @@ def is_integral_domain(self, proof=True):
                 True
                 sage: L.is_integral_domain(proof=True)                                  # needs sage.combinat
                 True
+
+                sage: ZZ['x'].is_integral_domain()
+                True
             """
             return True
 

src/sage/categories/modules.py

@@ -885,7 +885,8 @@ def __init_extra__(self):
                     sage: M.category()                                                  # needs sage.modules
                     Category of Cartesian products of modules with basis
                      over (Dedekind domains and euclidean domains
-                     and infinite enumerated sets and metric spaces)
+                      and noetherian rings
+                      and infinite enumerated sets and metric spaces)
                     sage: M.base_ring()                                                 # needs sage.modules
                     Integer Ring
 

src/sage/categories/morphism.pyx

@@ -179,9 +179,12 @@ cdef class Morphism(Map):
             sage: f.category()
             Category of endsets of unital magmas and right modules over
              (Dedekind domains and euclidean domains
+              and noetherian rings
+              and infinite enumerated sets and metric spaces)
+             and left modules over
+             (Dedekind domains and euclidean domains
+              and noetherian rings
               and infinite enumerated sets and metric spaces)
-             and left modules over (Dedekind domains and euclidean domains
-             and infinite enumerated sets and metric spaces)
 
             sage: # needs sage.rings.number_field
             sage: K = CyclotomicField(12)