gh-39497: moving random_element to category of rings
    
moving the random_element out of the auld `Ring` class

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #39497
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -132,7 +132,6 @@ This base class provides a lot more methods than a general parent::
      'ngens',
      'one',
      'order',
-     'random_element',
      'zero',
      'zeta',
      'zeta_order']

src/sage/categories/rings.py

@@ -17,6 +17,7 @@
 
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_import import LazyImport
+from sage.misc.prandom import randint
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.rngs import Rngs
 from sage.structure.element import Element
@@ -1529,6 +1530,40 @@ def _random_nonzero_element(self, *args, **kwds):
                 if not x.is_zero():
                     return x
 
+        def random_element(self, *args):
+            """
+            Return a random integer coerced into this ring.
+
+            INPUT:
+
+            - either no integer, one integer or two integers
+
+            The integer is chosen uniformly from the closed interval
+            ``[-2,2]``, ``[-a,a]`` or ``[a,b]`` according to the
+            length of the input.
+
+            ALGORITHM:
+
+            This uses Python's ``randint``.
+
+            EXAMPLES::
+
+                sage: -8 <= ZZ.random_element(8) <= 8
+                True
+                sage: -8 <= QQ.random_element(8) <= 8
+                True
+                sage: 4 <= ZZ.random_element(4,12) <= 12
+                True
+            """
+            if not args:
+                a, b = -2, 2
+            elif len(args) == 1:
+                bound = args[0]
+                a, b = -bound, bound
+            else:
+                a, b = args[0], args[1]
+            return randint(a, b) * self.one()
+
     class ElementMethods:
         def is_unit(self) -> bool:
             r"""

src/sage/categories/sets_cat.py

@@ -2472,11 +2472,11 @@ def random_element(self, *args):
                     sage: c2 = C.random_element(4,7)
                     sage: c2                   # random
                     (6, 5, 6, 4, 5, 6, 6, 4, 5, 5)
-                    sage: all(4 <= i < 7 for i in c2)
+                    sage: all(4 <= i <= 7 for i in c2)
                     True
                 """
                 return self._cartesian_product_of_elements(
-                        c.random_element(*args) for c in self.cartesian_factors())
+                    c.random_element(*args) for c in self.cartesian_factors())
 
             @abstract_method
             def _sets_keys(self):

src/sage/rings/finite_rings/integer_mod_ring.py

@@ -67,7 +67,7 @@
 from sage.arith.misc import factor
 from sage.arith.misc import primitive_root
 from sage.arith.misc import CRT_basis
-from sage.rings.ring import Field, CommutativeRing
+from sage.rings.ring import Field
 from sage.misc.mrange import cartesian_product_iterator
 import sage.rings.abc
 from sage.rings.finite_rings import integer_mod
@@ -1541,14 +1541,15 @@ def random_element(self, bound=None):
             sage: while not all(found):
             ....:     found[R.random_element()] = True
 
-        We test ``bound``-option::
+        We test the ``bound`` option::
 
-            sage: R.random_element(2) in [R(16), R(17), R(0), R(1), R(2)]
+            sage: R.random_element(2) in [R(-2), R(-1), R(0), R(1), R(2)]
             True
         """
         if bound is not None:
-            return CommutativeRing.random_element(self, bound)
-        a = random.randint(0, self.order() - 1)
+            a = random.randint(-bound, bound)
+        else:
+            a = random.randint(0, self.order() - 1)
         return self(a)
 
     @staticmethod

src/sage/rings/ring.pyx

@@ -114,7 +114,6 @@ from sage.misc.superseded import deprecation
 from sage.structure.coerce cimport coercion_model
 from sage.structure.parent cimport Parent
 from sage.structure.category_object cimport check_default_category
-from sage.misc.prandom import randint
 from sage.categories.rings import Rings
 from sage.categories.algebras import Algebras
 from sage.categories.commutative_algebras import CommutativeAlgebras
@@ -637,36 +636,6 @@ cdef class Ring(ParentWithGens):
         """
         return self.zeta().multiplicative_order()
 
-    def random_element(self, bound=2):
-        """
-        Return a random integer coerced into this ring, where the
-        integer is chosen uniformly from the interval ``[-bound,bound]``.
-
-        INPUT:
-
-        - ``bound`` -- integer (default: 2)
-
-        ALGORITHM:
-
-        Uses Python's randint.
-
-        TESTS:
-
-        The following example returns a :exc:`NotImplementedError` since the
-        generic ring class ``__call__`` function returns a
-        :exc:`NotImplementedError`. Note that
-        ``sage.rings.ring.Ring.random_element`` performs a call in the generic
-        ring class by a random integer::
-
-            sage: R = sage.rings.ring.Ring(ZZ); R
-            <sage.rings.ring.Ring object at ...>
-            sage: R.random_element()
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: cannot construct elements of <sage.rings.ring.Ring object at ...>
-        """
-        return self(randint(-bound,bound))
-
     @cached_method
     def epsilon(self):
         """