gh-37564: use Parent in infinity
    
Let us use `Parent` for the ugly Unsigned Infinity Ring.

The case of the Infinity Ring seems to be even more problematic, and is
not touched upon here.

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #37564
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/rings/infinity.py

@@ -219,7 +219,9 @@
 
 import sage.rings.abc
 
+from sage.structure.parent import Parent
 from sage.categories.rings import Rings
+from sage.categories.semirings import Semirings
 from sage.misc.fast_methods import Singleton
 from sage.misc.lazy_import import lazy_import
 from sage.rings.ring import CommutativeRing
@@ -232,7 +234,7 @@
 _obj = {}
 
 
-class _uniq():
+class _uniq:
     def __new__(cls, *args):
         """
         This ensures uniqueness of these objects.
@@ -248,7 +250,7 @@ def __new__(cls, *args):
         return O
 
 
-class AnInfinity():
+class AnInfinity:
     """
     TESTS::
 
@@ -560,7 +562,7 @@ def _sage_input_(self, sib, coerced):
             return -sib.name('oo')
 
 
-class UnsignedInfinityRing_class(Singleton, CommutativeRing):
+class UnsignedInfinityRing_class(Singleton, Parent):
 
     def __init__(self):
         """
@@ -584,7 +586,9 @@ def __init__(self):
             sage: UnsignedInfinityRing(3) == UnsignedInfinityRing(-19.5)
             True
         """
-        CommutativeRing.__init__(self, self, names=('oo',), normalize=False)
+        cat = Semirings().Commutative()
+        Parent.__init__(self, self, names=('oo',), normalize=False,
+                        category=cat)
 
     def ngens(self) -> int:
         """
@@ -599,19 +603,6 @@ def ngens(self) -> int:
         """
         return 1
 
-    def fraction_field(self):
-        """
-        The unsigned infinity ring isn't an integral domain.
-
-        EXAMPLES::
-
-            sage: UnsignedInfinityRing.fraction_field()
-            Traceback (most recent call last):
-            ...
-            TypeError: infinity 'ring' has no fraction field
-        """
-        raise TypeError("infinity 'ring' has no fraction field")
-
     def gen(self, n=0):
         """
         The "generator" of ``self`` is the infinity object.
@@ -1869,7 +1860,7 @@ def test_signed_infinity(pos_inf):
         sage: test_signed_infinity(RIF(oo))                                             # needs sage.rings.real_interval_field
         sage: test_signed_infinity(SR(oo))                                              # needs sage.symbolic
     """
-    msg = 'testing {} ({})'.format(pos_inf, type(pos_inf))
+    msg = f'testing {pos_inf} ({type(pos_inf)})'
     assert InfinityRing(pos_inf) is infinity, msg
     assert InfinityRing(-pos_inf) is minus_infinity, msg
     assert infinity == pos_inf, msg