gh-39133: Even less integral domain
    
remove two uses of the auld class `IntegralDomain`

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #39133
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

Author: Release Manager

src/sage/rings/polynomial/multi_polynomial_ring.py

@@ -60,14 +60,12 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from sage.rings.ring import IntegralDomain
 import sage.rings.fraction_field_element as fraction_field_element
 
 from sage.rings.polynomial.multi_polynomial_ring_base import MPolynomialRing_base, is_MPolynomialRing
 from sage.rings.polynomial.polynomial_singular_interface import PolynomialRing_singular_repr
 from sage.rings.polynomial.polydict import PolyDict, ETuple
 from sage.rings.polynomial.term_order import TermOrder
-
 import sage.interfaces.abc
 
 try:
@@ -558,9 +556,9 @@ def __call__(self, x=0, check=True):
         c = self.base_ring()(x)
         return MPolynomial_polydict(self, {self._zero_tuple: c})
 
-# The following methods are handy for implementing Groebner
-# basis algorithms. They do only superficial type/sanity checks
-# and should be called carefully.
+    # The following methods are handy for implementing Groebner
+    # basis algorithms. They do only superficial type/sanity checks
+    # and should be called carefully.
 
     def monomial_quotient(self, f, g, coeff=False):
         r"""
@@ -930,21 +928,17 @@ def sum(self, terms):
         elt = PolyDict({}, check=False)
         for t in terms:
             elt += self(t).element()
-        # NOTE: here we should be using self.element_class but polynomial rings are not complient
-        # with categories...
+        # NOTE: here we should be using self.element_class but
+        # polynomial rings are not complient with categories...
         from sage.rings.polynomial.multi_polynomial_element import MPolynomial_polydict
         return MPolynomial_polydict(self, elt)
 
 
-class MPolynomialRing_polydict_domain(IntegralDomain,
-                                      MPolynomialRing_polydict):
+class MPolynomialRing_polydict_domain(MPolynomialRing_polydict):
     def __init__(self, base_ring, n, names, order):
         order = TermOrder(order, n)
         MPolynomialRing_polydict.__init__(self, base_ring, n, names, order)
 
-    def is_integral_domain(self, proof=True):
-        return True
-
     def is_field(self, proof=True):
         if self.ngens() == 0:
             return self.base_ring().is_field(proof)

src/sage/rings/polynomial/polynomial_ring.py

@@ -150,7 +150,7 @@
 from sage.categories.principal_ideal_domains import PrincipalIdealDomains
 from sage.categories.rings import Rings
 
-from sage.rings.ring import (Ring, IntegralDomain)
+from sage.rings.ring import Ring, CommutativeRing
 from sage.structure.element import RingElement
 import sage.rings.rational_field as rational_field
 from sage.rings.rational_field import QQ
@@ -1928,8 +1928,7 @@ def _roots_univariate_polynomial(self, p, ring=None, multiplicities=True, algori
         return roots
 
 
-class PolynomialRing_integral_domain(PolynomialRing_commutative, PolynomialRing_singular_repr,
-                                     IntegralDomain):
+class PolynomialRing_integral_domain(PolynomialRing_commutative, PolynomialRing_singular_repr, CommutativeRing):
     def __init__(self, base_ring, name='x', sparse=False, implementation=None,
             element_class=None, category=None):
         """