sagemathgh-36684: get rid of some ParentWithGens in real and complex …

…fields

    
another step towards suppression of `ParentWithGens`, here in real and
complex rings

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#36684
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe

src/sage/rings/complex_double.pyx

@@ -81,10 +81,10 @@ 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_gens import ParentWithGens
 from sage.structure.richcmp cimport rich_to_bool
 from sage.categories.morphism cimport Morphism
 from sage.structure.coerce cimport is_numpy_type
@@ -149,7 +149,7 @@ cdef class ComplexDoubleField_class(sage.rings.abc.ComplexDoubleField):
             (-1.0, -1.0 + 1.2246...e-16*I, False)
         """
         from sage.categories.fields import Fields
-        ParentWithGens.__init__(self, self, ('I',), normalize=False, category=Fields().Infinite().Metric().Complete())
+        Ring.__init__(self, self, ('I',), normalize=False, category=Fields().Infinite().Metric().Complete())
         self._populate_coercion_lists_()
 
     def __reduce__(self):

src/sage/rings/complex_mpc.pyx

@@ -68,11 +68,11 @@ from sage.cpython.string cimport str_to_bytes
 from sage.libs.mpfr cimport *
 from sage.libs.mpc cimport *
 from sage.structure.parent cimport Parent
-from sage.structure.parent_gens cimport ParentWithGens
 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 +319,7 @@ 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)])
 
-        ParentWithGens.__init__(self, self._real_field(), ('I',), False, category=Fields().Infinite())
+        Ring.__init__(self, self._real_field(), ('I',), 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

@@ -40,7 +40,7 @@ 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.structure.parent_gens import ParentWithGens
+from sage.rings.ring import Ring
 
 from sage.misc.sage_eval import sage_eval
 
@@ -277,7 +277,7 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         """
         self._prec = int(prec)
         from sage.categories.fields import Fields
-        ParentWithGens.__init__(self, self._real_field(), ('I',), False, category=Fields().Infinite().Metric().Complete())
+        Ring.__init__(self, self._real_field(), ('I',), 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

@@ -107,15 +107,15 @@ Make sure we don't have a new field for every new literal::
     True
 """
 
-#*****************************************************************************
+# *****************************************************************************
 #       Copyright (C) 2005-2006 William Stein <wstein@gmail.com>
 #
 # 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.
 #                  https://www.gnu.org/licenses/
-#*****************************************************************************
+# *****************************************************************************
 
 import math # for log
 import sys
@@ -158,8 +158,8 @@ from sage.rings.real_double cimport RealDoubleElement
 import sage.rings.rational_field
 
 import sage.rings.infinity
+from sage.rings.ring import Ring
 
-from sage.structure.parent_gens cimport ParentWithGens
 from sage.arith.numerical_approx cimport digits_to_bits
 from sage.arith.constants cimport M_LN2_LN10
 from sage.arith.long cimport is_small_python_int
@@ -169,11 +169,11 @@ cimport gmpy2
 gmpy2.import_gmpy2()
 
 
-#*****************************************************************************
+# ****************************************************************************
 #
 #       Implementation
 #
-#*****************************************************************************
+# ****************************************************************************
 
 _re_skip_zeroes = re.compile(r'^(.+?)0*$')
 
@@ -356,11 +356,11 @@ def mpfr_get_exp_max_max():
 mpfr_set_exp_min(mpfr_get_emin_min())
 mpfr_set_exp_max(mpfr_get_emax_max())
 
-#*****************************************************************************
+# *****************************************************************************
 #
 #       Real Field
 #
-#*****************************************************************************
+# *****************************************************************************
 # The real field is in Cython, so mpfr elements will have access to
 # their parent via direct C calls, which will be faster.
 
@@ -534,7 +534,7 @@ 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
-        ParentWithGens.__init__(self, self, tuple(), False, category=Fields().Infinite().Metric().Complete())
+        Ring.__init__(self, self, tuple(), False, category=Fields().Infinite().Metric().Complete())
 
         # Initialize zero and one
         cdef RealNumber rn