gh-37234: make a category of Dedekind domains, remove code from ring.pyx
    
This is creating a `DedekindDomains` category and moving the associated
code from `ring.pyx` to the category.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [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.
- [x] I have updated the documentation accordingly.
    
URL: #37234
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/doc/en/tutorial/tour_coercion.rst

@@ -116,7 +116,8 @@ implemented in Sage as well:
     sage: Rings()
     Category of rings
     sage: ZZ.category()
-    Join of Category of euclidean domains
+    Join of Category of Dedekind domains
+        and Category of euclidean domains
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/doc/fr/tutorial/tour_coercion.rst

@@ -117,7 +117,8 @@ par ailleurs les catégories en tant que telles :
     sage: Rings()
     Category of rings
     sage: ZZ.category()
-    Join of Category of euclidean domains
+    Join of Category of Dedekind domains
+        and Category of euclidean domains
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/doc/ja/tutorial/tour_coercion.rst

@@ -99,7 +99,8 @@ 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: Rings()
     Category of rings
     sage: ZZ.category()
-    Join of Category of euclidean domains
+    Join of Category of Dedekind domains
+        and Category of euclidean domains
         and Category of infinite enumerated sets
         and Category of metric spaces
     sage: ZZ.category().is_subcategory(Rings())

src/doc/pt/tutorial/tour_coercion.rst

@@ -123,10 +123,10 @@ categorias matemáticas também são implementadas no Sage:
     sage: Rings()
     Category of rings
     sage: ZZ.category()
-    Join of Category of euclidean domains
+    Join of Category of Dedekind domains
+        and Category of euclidean domains
         and Category of infinite enumerated sets
         and Category of metric spaces
-
     sage: ZZ.category().is_subcategory(Rings())
     True
     sage: ZZ in Rings()

src/sage/algebras/clifford_algebra.py

@@ -416,7 +416,8 @@ def __init__(self, Q, names, category=None):
             sage: Cl = CliffordAlgebra(Q)
             sage: Cl.category()
             Category of finite dimensional super algebras with basis over
-             (euclidean domains and infinite enumerated sets and metric spaces)
+             (Dedekind domains and euclidean domains
+              and infinite enumerated sets and metric spaces)
             sage: TestSuite(Cl).run()
 
         TESTS:
@@ -994,7 +995,8 @@ def lift_module_morphism(self, m, names=None):
             sage: phi = Cl.lift_module_morphism(m, 'abc')
             sage: phi.category_for()
             Category of finite dimensional super algebras with basis over
-             (euclidean domains and infinite enumerated sets and metric spaces)
+             (Dedekind domains and euclidean domains
+              and infinite enumerated sets and metric spaces)
             sage: phi.matrix()
             [  1   0   0   0   7  -3  -7   0]
             [  0   1  -1  -1   0   0   0 -17]
@@ -1077,7 +1079,8 @@ def lift_isometry(self, m, names=None):
             sage: phi = Cl.lift_isometry(m, 'abc')
             sage: phi.category_for()
             Category of finite dimensional super algebras with basis over
-             (euclidean domains and infinite enumerated sets and metric spaces)
+             (Dedekind domains and euclidean domains
+              and infinite enumerated sets and metric spaces)
             sage: phi.matrix()
             [ 1  0  0  0  1  2  5  0]
             [ 0  1  1  2  0  0  0  5]

src/sage/categories/all.py

@@ -81,7 +81,7 @@
 RingModules = Modules
 from sage.categories.vector_spaces import VectorSpaces
 
-# (hopf) algebra structures
+# (Hopf) algebra structures
 from sage.categories.algebras import Algebras
 from sage.categories.commutative_algebras import CommutativeAlgebras
 from sage.categories.coalgebras import Coalgebras

src/sage/categories/basic.py

@@ -42,6 +42,7 @@
 from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.integral_domains import IntegralDomains
 from sage.categories.gcd_domains import GcdDomains
+from sage.categories.dedekind_domains import DedekindDomains
 from sage.categories.principal_ideal_domains import PrincipalIdealDomains
 from sage.categories.euclidean_domains import EuclideanDomains
 from sage.categories.unique_factorization_domains import UniqueFactorizationDomains

src/sage/categories/bimodules.py

@@ -75,14 +75,16 @@ def _make_named_class_key(self, name):
             (Join of Category of number fields
                  and Category of quotient fields
                  and Category of metric spaces,
-             Join of Category of euclidean domains
+             Join of Category of Dedekind domains
+                 and Category of euclidean domains
                  and Category of infinite enumerated sets
                  and Category of metric spaces)
 
 
             sage: Bimodules(Fields(), ZZ)._make_named_class_key('element_class')
             (Category of fields,
-             Join of Category of euclidean domains
+             Join of Category of Dedekind domains
+             and Category of euclidean domains
              and Category of infinite enumerated sets
              and Category of metric spaces)
 

src/sage/categories/category.py

@@ -119,7 +119,7 @@
 _join_cache = WeakValueDictionary()
 
 
-HALL_OF_FAME = ['Coxeter', 'Hopf', 'Weyl', 'Lie', 'Hecke']
+HALL_OF_FAME = ['Coxeter', 'Hopf', 'Weyl', 'Lie', 'Hecke', 'Dedekind']
 
 
 class Category(UniqueRepresentation, SageObject):
@@ -2592,6 +2592,7 @@ def category_sample():
         sage: from sage.categories.category import category_sample
         sage: sorted(category_sample(), key=str)                                        # needs sage.groups
         [Category of Coxeter groups,
+         Category of Dedekind domains,
          Category of G-sets for Symmetric group of order 8! as a permutation group,
          Category of Hecke modules over Rational Field,
          Category of Hopf algebras over Rational Field,
@@ -2836,7 +2837,8 @@ def _make_named_class_key(self, name):
         The parent class of an algebra depends only on the category of the base ring::
 
             sage: Algebras(ZZ)._make_named_class_key("parent_class")
-            Join of Category of euclidean domains
+            Join of Category of Dedekind domains
+                 and Category of euclidean domains
                  and Category of infinite enumerated sets
                  and Category of metric spaces
 
@@ -2847,7 +2849,8 @@ def _make_named_class_key(self, name):
             (Join of Category of number fields
                  and Category of quotient fields
                  and Category of metric spaces,
-             Join of Category of euclidean domains
+             Join of Category of Dedekind domains
+                 and Category of euclidean domains
                  and Category of infinite enumerated sets
                  and Category of metric spaces)
 
@@ -2973,7 +2976,8 @@ def _make_named_class_key(self, name):
         EXAMPLES::
 
             sage: Modules(ZZ)._make_named_class_key('element_class')
-            Join of Category of euclidean domains
+            Join of Category of Dedekind domains
+                 and Category of euclidean domains
                  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

@@ -225,7 +225,8 @@ def _make_named_class_key(self, name):
         EXAMPLES::
 
             sage: Modules(ZZ)._make_named_class_key('element_class')
-            Join of Category of euclidean domains
+            Join of Category of Dedekind domains
+             and  Category of euclidean domains
              and Category of infinite enumerated sets
              and Category of metric spaces
             sage: Modules(QQ)._make_named_class_key('parent_class')