gh-36933: More conversion to parent
    
a few more conversions from `Ring` (old coercion framework) to `Parent`
(new coercion framework),

### 📝 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.
- [ ] I have updated the documentation accordingly.
    
URL: #36933
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/rings/asymptotic/asymptotics_multivariate_generating_functions.py

@@ -185,22 +185,23 @@
 Classes and Methods
 ===================
 """
-#*****************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2008 Alexander Raichev <[email protected]>
 #       Copyright (C) 2014, 2016 Daniel Krenn <[email protected]>
 #
 # This program is free software: you can redistribute it and/or modify
 # 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 functools import total_ordering
 from itertools import combinations_with_replacement
+
 from sage.structure.element import RingElement
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.rings.ring import Ring
+from sage.structure.parent import Parent
 from sage.calculus.var import var
 from sage.calculus.functional import diff
 from sage.symbolic.ring import SR
@@ -1507,7 +1508,7 @@ def asymptotic_decomposition(self, alpha, asy_var=None):
         for f in decomp2:
             ff = self.parent()((f.numerator() /
                                 cauchy_stuff).simplify_full().collect(asy_var),
-                      f.denominator_factored())
+                               f.denominator_factored())
             decomp3.append(ff)
 
         return decomp3
@@ -2248,9 +2249,9 @@ def asymptotics_multiple(self, p, alpha, N, asy_var, coordinate=None,
         if verbose:
             print("Computing derivatives of auxiliary functions...")
         m = min(n, N)
-        end = [X[d-1] for j in range(n)]
+        end = [X[d - 1] for j in range(n)]
         Hprodderivs = diff_all(Hprod, X, 2 * N - 2 + n, ending=end, sub_final=P)
-        atP.update({U.subs(P): diff(Hprod, X[d - 1], n).subs(P)/factorial(n)})
+        atP.update({U.subs(P): diff(Hprod, X[d - 1], n).subs(P) / factorial(n)})
         Uderivs = {}
         k = Hprod.polynomial(CC).degree() - n
         if k == 0:
@@ -2327,7 +2328,7 @@ def asymptotics_multiple(self, p, alpha, N, asy_var, coordinate=None,
                                 (-1) ** (q - j - k)
                                 for (j, k) in product(range(min(n - 1, q) + 1),
                                                       range(max(0, q - n),
-                                                             q + 1))
+                                                            q + 1))
                                 if j + k <= q])
                            for q in range(N)])
         chunk = chunk.subs(P).simplify()
@@ -2936,7 +2937,7 @@ def relative_error(self, approx, alpha, interval, exp_scale=Integer(1),
         alpha = vector(alpha)
         multi_indices = [r * alpha for r in interval]
         mac = self.maclaurin_coefficients(multi_indices, numerical=digits)
-        #mac = self.old_maclaurin_coefficients(alpha, max(interval))
+        # mac = self.old_maclaurin_coefficients(alpha, max(interval))
         mac_approx = {}
         stats = []
         for r in interval:
@@ -3010,7 +3011,7 @@ def _mul_(left, right):
         return left.parent()(numer, df)
 
 
-class FractionWithFactoredDenominatorRing(UniqueRepresentation, Ring):
+class FractionWithFactoredDenominatorRing(UniqueRepresentation, Parent):
     r"""
     This is the ring of fractions with factored denominator.
 
@@ -3053,7 +3054,8 @@ def __classcall_private__(cls, denominator_ring, numerator_ring=None, category=N
             sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
             sage: R.<x,y> = PolynomialRing(QQ)
             sage: FFPD1 = FractionWithFactoredDenominatorRing(R)
-            sage: FFPD2 = FractionWithFactoredDenominatorRing(R, R, Rings())
+            sage: cat = Rings().Commutative()
+            sage: FFPD2 = FractionWithFactoredDenominatorRing(R, R, cat)
             sage: FFPD1 is FFPD2
             True
         """
@@ -3063,7 +3065,7 @@ def __classcall_private__(cls, denominator_ring, numerator_ring=None, category=N
             raise ValueError('numerator ring {} has no coercion map from the '
                              'denominator ring {}'.format(
                                  numerator_ring, denominator_ring))
-        category = Rings().or_subcategory(category)
+        category = Rings().Commutative().or_subcategory(category)
         return super().__classcall__(cls, denominator_ring,
                                      numerator_ring, category)
 
@@ -3081,11 +3083,11 @@ def __init__(self, denominator_ring, numerator_ring=None, category=None):
         """
         self._numerator_ring = numerator_ring
         self._denominator_ring = denominator_ring
-        Ring.__init__(self, denominator_ring, category=category)
+        Parent.__init__(self, denominator_ring, category=category)
 
     def _repr_(self):
         r"""
-        Returns a representation.
+        Return a representation.
 
         OUTPUT:
 
@@ -3614,7 +3616,7 @@ def sum(self):
 
 
 #####################################################################
-## Helper functions
+#  Helper functions
 
 
 def diff_prod(f_derivs, u, g, X, interval, end, uderivs, atc):
@@ -3829,7 +3831,7 @@ def subs_all(f, sub, simplify=False):
 
 
 def diff_all(f, V, n, ending=[], sub=None, sub_final=None,
-              zero_order=0, rekey=None):
+             zero_order=0, rekey=None):
     r"""
     Return a dictionary of representative mixed partial
     derivatives of `f` from order 1 up to order `n` with respect to the

src/sage/rings/complex_double.pyx

@@ -81,10 +81,9 @@ cdef extern from "<complex.h>":
 import sage.rings.abc
 cimport sage.rings.ring
 cimport sage.rings.integer
-from sage.rings.ring import Ring
 
 from sage.structure.element cimport Element, FieldElement
-from sage.structure.parent  cimport Parent
+from sage.structure.parent cimport Parent
 from sage.structure.richcmp cimport rich_to_bool
 from sage.categories.morphism cimport Morphism
 from sage.structure.coerce cimport is_numpy_type
@@ -149,7 +148,8 @@ cdef class ComplexDoubleField_class(sage.rings.abc.ComplexDoubleField):
             (-1.0, -1.0 + 1.2246...e-16*I, False)
         """
         from sage.categories.fields import Fields
-        Ring.__init__(self, self, ('I',), normalize=False, category=Fields().Infinite().Metric().Complete())
+        Parent.__init__(self, self, names=('I',), normalize=False,
+                        category=Fields().Infinite().Metric().Complete())
         self._populate_coercion_lists_()
 
     def __reduce__(self):

src/sage/rings/complex_mpc.pyx

@@ -72,7 +72,6 @@ from sage.structure.element cimport Element
 from sage.structure.richcmp cimport rich_to_bool
 from sage.categories.map cimport Map
 from sage.libs.pari.all import pari
-from sage.rings.ring import Ring
 
 from sage.rings.integer cimport Integer
 from sage.rings.complex_mpfr cimport ComplexNumber
@@ -319,7 +318,8 @@ cdef class MPComplexField_class(sage.rings.ring.Field):
         self.__real_field = real_mpfr.RealField(prec, rnd=_mpfr_rounding_modes[rnd_re(n)])
         self.__imag_field = real_mpfr.RealField(prec, rnd=_mpfr_rounding_modes[rnd_im(n)])
 
-        Ring.__init__(self, self._real_field(), ('I',), False, category=Fields().Infinite())
+        Parent.__init__(self, self._real_field(), names=('I',), normalize=False,
+                        category=Fields().Infinite())
         self._populate_coercion_lists_(coerce_list=[MPFRtoMPC(self._real_field(), self)])
 
     cdef MPComplexNumber _new(self) noexcept:

src/sage/rings/complex_mpfr.pyx

@@ -36,11 +36,10 @@ import sage.misc.misc
 
 from sage.libs.mpfr cimport *
 
+from sage.structure.parent cimport Parent
 from sage.structure.element cimport RingElement, Element
 from sage.structure.richcmp cimport rich_to_bool
 from sage.categories.map cimport Map
-from sage.structure.parent import Parent
-from sage.rings.ring import Ring
 
 from sage.misc.sage_eval import sage_eval
 
@@ -282,7 +281,9 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         """
         self._prec = int(prec)
         from sage.categories.fields import Fields
-        Ring.__init__(self, self._real_field(), ('I',), False, category=Fields().Infinite().Metric().Complete())
+        Parent.__init__(self, self._real_field(), names=('I',),
+                        normalize=False,
+                        category=Fields().Infinite().Metric().Complete())
         self._populate_coercion_lists_(coerce_list=[RRtoCC(self._real_field(), self)],
                 convert_method_name='_complex_mpfr_')
 

src/sage/rings/real_mpfr.pyx

@@ -143,18 +143,20 @@ from sage.libs.mpfr cimport *
 from sage.libs.mpmath.utils cimport mpfr_to_mpfval
 from sage.misc.randstate cimport randstate, current_randstate
 from sage.misc.superseded import deprecation_cython as deprecation
-from sage.rings.integer cimport Integer
-from sage.rings.rational cimport Rational
-from sage.rings.real_double cimport RealDoubleElement
-from sage.rings.ring import Ring
+
 from sage.structure.element cimport Element
+from sage.structure.parent cimport Parent
 from sage.structure.element cimport have_same_parent
-from sage.structure.parent_gens cimport ParentWithGens
 from sage.structure.richcmp cimport rich_to_bool_sgn
-
 cdef bin_op
 from sage.structure.element import bin_op
 
+from sage.libs.mpmath.utils cimport mpfr_to_mpfval
+
+from sage.rings.integer cimport Integer
+from sage.rings.rational cimport Rational
+from sage.rings.real_double cimport RealDoubleElement
+
 try:
     from cypari2 import Gen
     from sage.libs.pari.convert_sage_real_mpfr import new_gen_from_real_mpfr_element, set_real_mpfr_element_from_gen
@@ -530,7 +532,8 @@ cdef class RealField_class(sage.rings.abc.RealField):
         self.rnd_str = char_to_str(rnd_str + 5)  # Strip "MPFR_"
 
         from sage.categories.fields import Fields
-        Ring.__init__(self, self, tuple(), False, category=Fields().Infinite().Metric().Complete())
+        Parent.__init__(self, self, names=tuple(), normalize=False,
+                        category=Fields().Infinite().Metric().Complete())
 
         # Initialize zero and one
         cdef RealNumber rn