gh-40160: get rid of CommutativeRing in p-adics
    
using `Parent` instead of the auld class `CommutativeRing` there

### 📝 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: #40160
Reported by: Frédéric Chapoton
Reviewer(s): Rubén Muñoz--Bertrand, Travis Scrimshaw

Author: Release Manager

src/sage/rings/padics/local_generic.py

@@ -27,10 +27,10 @@
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.rings.infinity import Infinity
-from sage.rings.ring import CommutativeRing
+from sage.structure.parent import Parent
 
 
-class LocalGeneric(CommutativeRing):
+class LocalGeneric(Parent):
     def __init__(self, base, prec, names, element_class, category=None):
         r"""
         Initialize ``self``.
@@ -74,10 +74,10 @@ def __init__(self, base, prec, names, element_class, category=None):
         category = category.Metric().Complete().Infinite()
         if default_category is not None:
             category = check_default_category(default_category, category)
-        CommutativeRing.__init__(self, base, names=(names,),
-                                 normalize=False, category=category)
+        Parent.__init__(self, base=base, names=(names,),
+                        normalize=False, category=category)
 
-    def is_capped_relative(self):
+    def is_capped_relative(self) -> bool:
         r"""
         Return whether this `p`-adic ring bounds precision in a capped
         relative fashion.
@@ -102,7 +102,7 @@ def is_capped_relative(self):
         """
         return False
 
-    def is_capped_absolute(self):
+    def is_capped_absolute(self) -> bool:
         r"""
         Return whether this `p`-adic ring bounds precision in a
         capped absolute fashion.
@@ -127,7 +127,7 @@ def is_capped_absolute(self):
         """
         return False
 
-    def is_fixed_mod(self):
+    def is_fixed_mod(self) -> bool:
         r"""
         Return whether this `p`-adic ring bounds precision in a fixed
         modulus fashion.
@@ -154,7 +154,7 @@ def is_fixed_mod(self):
         """
         return False
 
-    def is_floating_point(self):
+    def is_floating_point(self) -> bool:
         r"""
         Return whether this `p`-adic ring bounds precision in a floating
         point fashion.
@@ -179,7 +179,7 @@ def is_floating_point(self):
         """
         return False
 
-    def is_lattice_prec(self):
+    def is_lattice_prec(self) -> bool:
         r"""
         Return whether this `p`-adic ring bounds precision using
         a lattice model.
@@ -208,7 +208,7 @@ def is_lattice_prec(self):
         """
         return False
 
-    def is_relaxed(self):
+    def is_relaxed(self) -> bool:
         r"""
         Return whether this `p`-adic ring bounds precision in a relaxed
         fashion.
@@ -224,7 +224,7 @@ def is_relaxed(self):
         """
         return False
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Latex.
 

src/sage/rings/padics/witt_vector_ring.py

@@ -21,11 +21,11 @@
 # (at your option) any later version.
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
-
-
 from itertools import product
+from typing import Iterator
 
 from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.integral_domains import IntegralDomains
 from sage.categories.fields import Fields
 from sage.misc.latex import latex
 from sage.rings.integer import Integer
@@ -42,15 +42,14 @@
 from sage.rings.polynomial.polynomial_element import Polynomial
 from sage.rings.polynomial.polynomial_ring import PolynomialRing_generic
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.ring import CommutativeRing
+from sage.structure.parent import Parent
 from sage.sets.primes import Primes
 from sage.structure.unique_representation import UniqueRepresentation
 
 
 def fast_char_p_power(x, n, p=None):
     r"""
-    Return `x^n` assuming that `x` lives in a ring of
-    characteristic `p`.
+    Return `x^n` assuming that `x` lives in a ring of characteristic `p`.
 
     If `x` is not an element of a ring of characteristic `p`,
     this throws an error.
@@ -120,7 +119,7 @@ def fast_char_p_power(x, n, p=None):
     return xn
 
 
-class WittVectorRing(CommutativeRing, UniqueRepresentation):
+class WittVectorRing(Parent, UniqueRepresentation):
     r"""
     Return the appropriate `p`-typical truncated Witt vector ring.
 
@@ -263,27 +262,32 @@ def __classcall_private__(cls, coefficient_ring, prec=1, p=None, algorithm=None)
 
         return child.__classcall__(child, coefficient_ring, prec, p)
 
-    def __init__(self, coefficient_ring, prec, prime):
+    def __init__(self, coefficient_ring, prec, prime) -> None:
         r"""
-        Initialises ``self``.
+        Initialise ``self``.
 
         EXAMPLES::
 
-            sage: W = WittVectorRing(PolynomialRing(GF(5), 't'), prec=4)
-            sage: W
+            sage: W = WittVectorRing(PolynomialRing(GF(5), 't'), prec=4); W
             Ring of truncated 5-typical Witt vectors of length 4 over Univariate Polynomial Ring in t over Finite Field of size 5
             sage: type(W)
             <class 'sage.rings.padics.witt_vector_ring.WittVectorRing_phantom_with_category'>
 
             sage: TestSuite(W).run()
         """
-        self._coefficient_ring = coefficient_ring
+        cring = coefficient_ring
+        self._coefficient_ring = cring
         self._prec = prec
         self._prime = prime
 
-        CommutativeRing.__init__(self, self)
+        if prec == 1 and cring in IntegralDomains():
+            cat = IntegralDomains()
+        else:
+            cat = CommutativeRings()
 
-    def __iter__(self):
+        Parent.__init__(self, base=ZZ, category=cat)
+
+    def __iter__(self) -> Iterator:
         """
         Iterator for truncated Witt vector rings.
 
@@ -347,7 +351,7 @@ def _coerce_map_from_(self, S):
         if (isinstance(S, WittVectorRing)
             and S.precision() >= self._prec and S.prime() == self._prime
             and self._coefficient_ring.has_coerce_map_from(
-                     S.coefficient_ring())):
+                S.coefficient_ring())):
             return (any(isinstance(S, rng) for rng in self._always_coerce)
                     or (S.precision() != self._prec
                         or S.coefficient_ring() is not self._coefficient_ring)
@@ -358,7 +362,7 @@ def _coerce_map_from_(self, S):
 
     def _generate_sum_and_product_polynomials(self, coefficient_ring, prec, p):
         """
-        Generates the sum and product polynomials defining the ring laws of
+        Generate the sum and product polynomials defining the ring laws of
         truncated Witt vectors for the ``standard`` algorithm.
 
         EXAMPLES::
@@ -387,7 +391,7 @@ def _generate_sum_and_product_polynomials(self, coefficient_ring, prec, p):
         #
         # Remark: Since when is SIXTEEN bits sufficient for anyone???
         #
-        if p**(prec-1) >= 2**16:
+        if p**(prec - 1) >= 2**16:
             implementation = 'generic'
         else:
             implementation = 'singular'
@@ -421,7 +425,7 @@ def _generate_sum_and_product_polynomials(self, coefficient_ring, prec, p):
             self._sum_polynomials[n] = S(self._sum_polynomials[n])
             self._prod_polynomials[n] = S(self._prod_polynomials[n])
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a `\LaTeX` representation of ``self``.
 
@@ -439,7 +443,7 @@ def _latex_(self):
         return "W_{%s}\\left(%s\\right)" % (latex(self._prec),
                                             latex(self._coefficient_ring))
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of the ring.
 
@@ -498,7 +502,7 @@ def coefficient_ring(self):
         """
         return self._coefficient_ring
 
-    def is_finite(self):
+    def is_finite(self) -> bool:
         """
         Return whether ``self`` is a finite ring.
 
@@ -572,7 +576,9 @@ def prod_polynomials(self, variables=None):
 
     def random_element(self, *args, **kwds):
         """
-        Return a random truncated Witt vector. Extra arguments are passed to
+        Return a random truncated Witt vector.
+
+        Extra arguments are passed to
         the random generator of the coefficient ring.
 
         EXAMPLES::
@@ -621,8 +627,9 @@ def sum_polynomials(self, variables=None):
 
     def teichmuller_lift(self, x):
         """
-        Return the Teichmüller lift of ``x`` in ``self``. This lift is
-        sometimes known as the multiplicative lift of ``x``.
+        Return the Teichmüller lift of ``x`` in ``self``.
+
+        This lift is sometimes known as the multiplicative lift of ``x``.
 
         EXAMPLES::
 
@@ -656,9 +663,9 @@ class WittVectorRing_finotti(WittVectorRing):
     """
     Element = WittVector_finotti
 
-    def __init__(self, coefficient_ring, prec, prime):
+    def __init__(self, coefficient_ring, prec, prime) -> None:
         r"""
-        Initialises ``self``.
+        Initialise ``self``.
 
         EXAMPLES::
 
@@ -798,9 +805,9 @@ class WittVectorRing_phantom(WittVectorRing):
     """
     Element = WittVector_phantom
 
-    def __init__(self, coefficient_ring, prec, prime):
+    def __init__(self, coefficient_ring, prec, prime) -> None:
         r"""
-        Initialises ``self``.
+        Initialise ``self``.
 
         EXAMPLES::
 
@@ -859,9 +866,9 @@ class WittVectorRing_pinvertible(WittVectorRing):
     """
     Element = WittVector_pinvertible
 
-    def __init__(self, coefficient_ring, prec, prime):
+    def __init__(self, coefficient_ring, prec, prime) -> None:
         r"""
-        Initialises ``self``.
+        Initialise ``self``.
 
         EXAMPLES::
 
@@ -901,9 +908,9 @@ class WittVectorRing_standard(WittVectorRing):
     """
     Element = WittVector_standard
 
-    def __init__(self, coefficient_ring, prec, prime):
+    def __init__(self, coefficient_ring, prec, prime) -> None:
         r"""
-        Initialises ``self``.
+        Initialise ``self``.
 
         EXAMPLES::