gh-38215: Fix typo "sage.ring" in docstrings
    
The correct module path is "sage.rings" instead of "sage.ring".
In sage/modular/pollack_stevens/padic_lseries.py and in
sage/structure/factory.pyx the typo prevents doctests from being run
because the "needs" requirement canot be fulfilled.
In sage/rings/polynomial/padics/polynomial_padic.py the typo is merely
cosmetic.

See brief [discussion in Sage Support
Group](https://groups.google.com/g/sage-support/c/Msx0QJF7rpA)

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URL: #38215
Reported by: Jens-Erik Riedel
Reviewer(s): David Coudert, Julian Rüth

Author: Release Manager

src/sage/modular/pollack_stevens/padic_lseries.py

@@ -1,4 +1,4 @@
-# sage.doctest: needs sage.ring.padics
+# sage.doctest: needs sage.rings.padics
 r"""
 `p`-adic `L`-series attached to overconvergent eigensymbols
 
@@ -174,7 +174,7 @@ def __eq__(self, other):
 
             sage: E = EllipticCurve('11a')
             sage: L = E.padic_lseries(11,implementation="pollackstevens",precision=6) # long time
-            sage: L == loads(dumps(L)) # indirect doctest long time
+            sage: L == loads(dumps(L)) # indirect doctest, long time
             True
         """
         if not isinstance(other, pAdicLseries):

src/sage/rings/polynomial/padics/polynomial_padic.py

@@ -274,7 +274,7 @@ def root_field(self, names, check_irreducible=True, **kwds):
 
         * ``check_irreducible`` -- check whether the polynomial is irreducible
 
-        * ``kwds`` -- see :meth:`sage.ring.padics.padic_generic.pAdicGeneric.extension`
+        * ``kwds`` -- see :meth:`sage.rings.padics.padic_generic.pAdicGeneric.extension`
 
         EXAMPLES::
 

src/sage/structure/factory.pyx

@@ -520,7 +520,7 @@ cdef class UniqueFactory(SageObject):
         The ``GF`` factory used to have a custom :meth:`other_keys`
         method, but this was removed in :issue:`16934`::
 
-            sage: # needs sage.libs.linbox sage.ring.finite_rings
+            sage: # needs sage.libs.linbox sage.rings.finite_rings
             sage: key, _ = GF.create_key_and_extra_args(27, 'k'); key
             (27, ('k',), x^3 + 2*x + 1, 'givaro', 3, 3, True, None, 'poly', True, True, True)
             sage: K = GF.create_object(0, key); K