sagemathgh-39131: avoid some imports of rings.Ring
    
getting rid of some imports of `rings.Ring`

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#39131
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

Author: Release Manager

src/sage/graphs/chrompoly.pyx

@@ -31,7 +31,7 @@ from memory_allocator cimport MemoryAllocator
 from sage.libs.gmp.mpz cimport *
 from sage.rings.integer_ring import ZZ
 from sage.rings.integer cimport Integer
-from sage.rings.ring cimport Ring
+from sage.rings.ring cimport CommutativeRing
 from sage.rings.polynomial.polynomial_integer_dense_flint cimport Polynomial_integer_dense_flint
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 
@@ -436,7 +436,7 @@ def chromatic_polynomial_with_cache(G, cache=None):
         ...
         TypeError: parameter cache must be a dictionary or None
     """
-    cdef Ring R = PolynomialRing(ZZ, "x", implementation="FLINT")
+    cdef CommutativeRing R = PolynomialRing(ZZ, "x", implementation="FLINT")
     cdef Polynomial_integer_dense_flint one = R.one()
     cdef Polynomial_integer_dense_flint zero = R.zero()
     cdef Polynomial_integer_dense_flint x = R.gen()

src/sage/modular/pollack_stevens/distributions.py

@@ -39,14 +39,14 @@
 #                  https://www.gnu.org/licenses/
 # *************************************************************************
 
+from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.fields import Fields
 from sage.categories.modules import Modules
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_import import lazy_import
 from sage.modules.module import Module
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
-from sage.rings.ring import Ring
 from sage.structure.factory import UniqueFactory
 from sage.structure.parent import Parent
 from .sigma0 import _default_adjuster
@@ -281,8 +281,8 @@ def __init__(self, k, p=None, prec_cap=None, base=None, character=None,
             ...
             ValueError: p must be prime
         """
-        if not isinstance(base, Ring):
-            raise TypeError("base must be a ring")
+        if base not in CommutativeRings():
+            raise TypeError("base must be a commutative ring")
         # from sage.rings.padics.pow_computer import PowComputer
         # should eventually be the PowComputer on ZpCA once that uses longs.
         Dist, WeightKAction = get_dist_classes(p, prec_cap, base,
@@ -649,10 +649,13 @@ def __init__(self, k, base, character, adjuster, act_on_left, dettwist,
         if hasattr(base, 'prime'):
             p = base.prime()
         else:
-            p = ZZ(0)
-        OverconvergentDistributions_abstract.__init__(self, k, p, k + 1, base, character,
-                                        adjuster, act_on_left, dettwist,
-                                        act_padic, implementation)
+            p = ZZ.zero()
+        OverconvergentDistributions_abstract.__init__(self, k, p, k + 1,
+                                                      base, character,
+                                                      adjuster, act_on_left,
+                                                      dettwist,
+                                                      act_padic,
+                                                      implementation)
 
     def _an_element_(self):
         r"""

src/sage/rings/number_field/number_field.py

@@ -75,10 +75,12 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 from __future__ import annotations
+from itertools import count
+from collections import Counter
+
 from sage.misc.cachefunc import cached_method
 from sage.misc.superseded import deprecation
 
-
 import sage.libs.ntl.all as ntl
 import sage.rings.abc
 import sage.rings.complex_mpfr
@@ -99,7 +101,6 @@
 from sage.rings.finite_rings.integer_mod import mod
 from sage.categories.number_fields import NumberFields
 
-from sage.rings.ring import Ring
 from sage.misc.latex import latex_variable_name
 
 from .unit_group import UnitGroup
@@ -118,8 +119,6 @@
 from . import maps
 from . import structure
 from . import number_field_morphisms
-from itertools import count
-from collections import Counter
 
 from sage.categories.homset import Hom
 from sage.categories.sets_cat import Sets
@@ -128,6 +127,7 @@
 from sage.rings.real_mpfr import RR
 
 from sage.interfaces.abc import GapElement
+from sage.rings.number_field.morphism import RelativeNumberFieldHomomorphism_from_abs
 
 lazy_import('sage.libs.gap.element', 'GapElement', as_='LibGapElement')
 lazy_import('sage.rings.universal_cyclotomic_field', 'UniversalCyclotomicFieldElement')
@@ -136,13 +136,12 @@
 _NumberFields = NumberFields()
 
 
-from sage.rings.number_field.morphism import RelativeNumberFieldHomomorphism_from_abs
-
-
 def is_NumberFieldHomsetCodomain(codomain):
     """
     Return whether ``codomain`` is a valid codomain for a number
-    field homset. This is used by NumberField._Hom_ to determine
+    field homset.
+
+    This is used by NumberField._Hom_ to determine
     whether the created homsets should be a
     :class:`sage.rings.number_field.homset.NumberFieldHomset`.
 
@@ -414,8 +413,8 @@ def NumberField(polynomial, name=None, check=True, names=None, embedding=None,
         sage: RR(g)
         -1.25992104989487
 
-    If no embedding is specified or is complex, the comparison is not returning something
-    meaningful.::
+    If no embedding is specified or is complex, the comparison is not
+    returning something meaningful. ::
 
         sage: N.<g> = NumberField(x^3 + 2)
         sage: 1 < g