gh-39227: deprecate fully the auld class IntegralDomain
    
as another little step towards removing the auld coercion framework

### 📝 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: #39227
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

Author: Release Manager

src/sage/rings/polynomial/polynomial_quotient_ring.py

@@ -34,15 +34,15 @@
 # it under the terms of the GNU General Public License as published by
 # the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
-#                  http://www.gnu.org/licenses/
+#                  https://www.gnu.org/licenses/
 #*****************************************************************************
 
 
 from . import polynomial_element
 import sage.rings.rational_field
 
 from sage.arith.misc import crt
-from sage.rings.ring import Field, IntegralDomain, CommutativeRing
+from sage.rings.ring import Field, CommutativeRing
 
 from sage.misc.cachefunc import cached_method
 from sage.rings.polynomial.polynomial_quotient_ring_element import PolynomialQuotientRingElement
@@ -2277,7 +2277,7 @@ def _richcmp_(self, other, op):
         return richcmp(self.parent(), other.parent(), op)
 
 
-class PolynomialQuotientRing_domain(PolynomialQuotientRing_generic, IntegralDomain):
+class PolynomialQuotientRing_domain(PolynomialQuotientRing_generic, CommutativeRing):
     """
     EXAMPLES::
 

src/sage/rings/ring.pyx

@@ -23,7 +23,7 @@ The class inheritance hierarchy is:
 
     - :class:`NoetherianRing` (deprecated and essentially removed)
     - :class:`CommutativeAlgebra` (deprecated and essentially removed)
-    - :class:`IntegralDomain` (deprecated)
+    - :class:`IntegralDomain` (deprecated and essentially removed)
 
       - :class:`DedekindDomain` (deprecated and essentially removed)
       - :class:`PrincipalIdealDomain` (deprecated and essentially removed)
@@ -954,50 +954,11 @@ cdef class CommutativeRing(Ring):
 
 
 cdef class IntegralDomain(CommutativeRing):
-    """
-    Generic integral domain class.
-
-    This class is deprecated. Please use the
-    :class:`sage.categories.integral_domains.IntegralDomains`
-    category instead.
-    """
     _default_category = IntegralDomains()
 
-    def __init__(self, base_ring, names=None, normalize=True, category=None):
-        """
-        Initialize ``self``.
-
-        INPUT:
-
-         - ``category`` -- (default: ``None``) a category, or ``None``
-
-        This method is used by all the abstract subclasses of
-        :class:`IntegralDomain`, like :class:`Field`, ... in order to
-        avoid cascade calls Field.__init__ ->
-        IntegralDomain.__init__ ->
-        ...
-
-        EXAMPLES::
-
-            sage: F = IntegralDomain(QQ)
-            sage: F.category()
-            Category of integral domains
-
-            sage: F = Field(QQ)
-            sage: F.category()
-            Category of fields
-
-        The default value for the category is specified by the class
-        attribute ``default_category``::
-
-            sage: IntegralDomain._default_category
-            Category of integral domains
-
-            sage: Field._default_category
-            Category of fields
-        """
-        CommutativeRing.__init__(self, base_ring, names=names, normalize=normalize,
-                                 category=category)
+    def __init__(self, *args, **kwds):
+        deprecation(39227, "use the category IntegralDomains")
+        super().__init__(*args, **kwds)
 
 
 cdef class NoetherianRing(CommutativeRing):