gh-39619: remove unused imports in rings
    
found using `ruff check --select=F401 src/sage/rings`

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #39619
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

Author: Release Manager

src/sage/rings/finite_rings/finite_field_constructor.py

@@ -573,8 +573,6 @@ def create_key_and_extra_args(self, order, name=None, modulus=None, names=None,
             ...
             ValueError: wrong input for finite field constructor
         """
-        import sage.arith.all
-
         for key, val in kwds.items():
             if key not in ['structure', 'implementation', 'prec', 'embedding', 'latex_names']:
                 raise TypeError("create_key_and_extra_args() got an unexpected keyword argument '%s'" % key)

src/sage/rings/function_field/drinfeld_modules/homset.py

@@ -17,7 +17,7 @@
 #  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/
 # *****************************************************************************
 
 import operator
@@ -27,7 +27,6 @@
 from sage.categories.action import Action
 from sage.misc.latex import latex
 from sage.rings.function_field.drinfeld_modules.morphism import DrinfeldModuleMorphism
-from sage.structure.parent import Parent
 
 
 class DrinfeldModuleMorphismAction(Action):

src/sage/rings/number_field/order_ideal.py

@@ -774,9 +774,8 @@ def _random_for_testing():
     from sage.rings.number_field.number_field import QuadraticField
     from sage.arith.misc import primes
     from sage.rings.finite_rings.integer_mod_ring import Zmod
-    from sage.misc.misc_c import prod
     while True:
-        d = ZZ(choice((-1,+1)) * randrange(1,10**5))
+        d = ZZ(choice((-1, +1)) * randrange(1, 10**5))
         if not d.is_square():
             break
     K,t = QuadraticField(d).objgen()

src/sage/rings/polynomial/multi_polynomial_element.py

@@ -68,7 +68,6 @@
 from sage.rings.polynomial.polynomial_singular_interface import Polynomial_singular_repr
 from sage.structure.sequence import Sequence
 from .multi_polynomial import MPolynomial, is_MPolynomial
-from sage.categories.morphism import Morphism
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.superseded import deprecated_function_alias
 from sage.rings.rational_field import QQ

src/sage/rings/polynomial/multi_polynomial_ideal.py

@@ -249,7 +249,6 @@
 from sage.rings.integer_ring import ZZ
 from sage.rings.noncommutative_ideals import Ideal_nc
 from sage.rings.qqbar_decorators import handle_AA_and_QQbar
-from sage.structure.element import parent
 from sage.structure.richcmp import (op_EQ, op_GE, op_GT, op_LE, op_LT, op_NE,
                                     rich_to_bool, richcmp_method)
 from sage.structure.sequence import Sequence, Sequence_generic

src/sage/rings/polynomial/polynomial_ring.py

@@ -166,7 +166,6 @@
 from sage.rings.polynomial.polynomial_ring_constructor import polynomial_default_category
 
 import sage.misc.latex as latex
-from sage.misc.prandom import randint
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_attribute import lazy_attribute
 

src/sage/rings/polynomial/polynomial_singular_interface.py

@@ -38,7 +38,6 @@
 
 import sage.rings.fraction_field
 import sage.rings.abc
-import sage.rings.number_field as number_field
 
 from sage.rings.rational_field import RationalField
 from sage.rings.function_field.function_field_rational import RationalFunctionField

src/sage/rings/power_series_ring.py

@@ -143,7 +143,6 @@
     power_series_mpoly,
     power_series_poly,
     power_series_ring_element,
-    ring,
 )
 from sage.rings.fraction_field_element import FractionFieldElement
 from sage.rings.infinity import infinity