gh-39232: minor tweaks in Ore polynomials
    
a few minor changes in Ore polynomials

- sort imports
- better categories
- no custom "is_commutative"

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #39232
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert

Author: Release Manager

src/sage/rings/polynomial/ore_function_element.py

@@ -16,14 +16,13 @@
 #                  https://www.gnu.org/licenses/
 # ***************************************************************************
 
-from sage.structure.richcmp import richcmp, op_EQ, op_NE
-from sage.misc.cachefunc import cached_method
-from sage.misc.latex import latex
-
+from sage.categories.homset import Hom
 from sage.categories.map import Map
 from sage.categories.morphism import Morphism
-from sage.categories.homset import Hom
+from sage.misc.cachefunc import cached_method
+from sage.misc.latex import latex
 from sage.structure.element import AlgebraElement
+from sage.structure.richcmp import op_EQ, op_NE, richcmp
 
 
 class OreFunction(AlgebraElement):

src/sage/rings/polynomial/ore_function_field.py

@@ -128,6 +128,23 @@
     sage: g^(-1) * f
     (d - 1/t)^(-1) * (d + (t^2 - 1)/t)
 
+TESTS:
+
+The Ore function field is commutative if the twisting morphism is the
+identity and the twisting derivation vanishes. ::
+
+    sage: # needs sage.rings.finite_rings
+    sage: k.<a> = GF(5^3)
+    sage: Frob = k.frobenius_endomorphism()
+    sage: S.<x> = k['x', Frob]
+    sage: K = S.fraction_field()
+    sage: K.is_commutative()
+    False
+    sage: T.<y> = k['y', Frob^3]
+    sage: L = T.fraction_field()
+    sage: L.is_commutative()
+    True
+
 AUTHOR:
 
 - Xavier Caruso (2020-05)
@@ -144,21 +161,21 @@
 #                  https://www.gnu.org/licenses/
 # ***************************************************************************
 
-import sage
-
-from sage.structure.richcmp import op_EQ
-from sage.structure.category_object import normalize_names
-from sage.structure.unique_representation import UniqueRepresentation
-from sage.structure.parent import Parent
 from sage.categories.algebras import Algebras
+from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.fields import Fields
-
-from sage.rings.morphism import RingHomomorphism
 from sage.categories.homset import Hom
 from sage.categories.map import Section
-
+from sage.rings.morphism import RingHomomorphism
+from sage.rings.polynomial.ore_function_element import (
+    OreFunction_with_large_center,
+    OreFunctionBaseringInjection,
+)
 from sage.rings.polynomial.ore_polynomial_ring import OrePolynomialRing
-from sage.rings.polynomial.ore_function_element import OreFunctionBaseringInjection
+from sage.structure.category_object import normalize_names
+from sage.structure.parent import Parent
+from sage.structure.richcmp import op_EQ
+from sage.structure.unique_representation import UniqueRepresentation
 
 WORKING_CENTER_MAX_TRIES = 1000
 
@@ -200,7 +217,10 @@ def __init__(self, ring, category=None):
             self._simplification = False
         self._ring = ring
         base = ring.base_ring()
-        category = Algebras(base).or_subcategory(category)
+        if ring in CommutativeRings():
+            category = Algebras(base).Commutative().or_subcategory(category)
+        else:
+            category = Algebras(base).or_subcategory(category)
         Parent.__init__(self, base=base, names=ring.variable_name(),
                         normalize=True, category=category)
 
@@ -614,27 +634,6 @@ def random_element(self, degree=2, monic=False, *args, **kwds):
         denominator = self._ring.random_element(degdenom, True, *args, **kwds)
         return self(numerator, denominator)
 
-    def is_commutative(self) -> bool:
-        r"""
-        Return ``True`` if this Ore function field is commutative, i.e. if the
-        twisting morphism is the identity and the twisting derivation vanishes.
-
-        EXAMPLES::
-
-            sage: # needs sage.rings.finite_rings
-            sage: k.<a> = GF(5^3)
-            sage: Frob = k.frobenius_endomorphism()
-            sage: S.<x> = k['x', Frob]
-            sage: K = S.fraction_field()
-            sage: K.is_commutative()
-            False
-            sage: T.<y> = k['y', Frob^3]
-            sage: L = T.fraction_field()
-            sage: L.is_commutative()
-            True
-        """
-        return self._ring.is_commutative()
-
     def is_field(self, proof=False) -> bool:
         r"""
         Return always ``True`` since Ore function field are (skew) fields.
@@ -892,7 +891,7 @@ def __init__(self, ring, category=None):
             sage: TestSuite(K).run()
         """
         if self.Element is None:
-            self.Element = sage.rings.polynomial.ore_function_element.OreFunction_with_large_center
+            self.Element = OreFunction_with_large_center
         OreFunctionField.__init__(self, ring, category)
         self._center = {}
         self._center_variable_name = 'z'

src/sage/rings/polynomial/ore_polynomial_ring.py

@@ -7,6 +7,30 @@
 which constructs a general dense univariate Ore polynomial ring over a
 commutative base with equipped with an endomorphism and/or a derivation.
 
+TESTS:
+
+The Ore polynomial ring is commutative if the twisting morphism is the
+identity and the twisting derivation vanishes. ::
+
+    sage: # needs sage.rings.finite_rings
+    sage: k.<a> = GF(5^3)
+    sage: Frob = k.frobenius_endomorphism()
+    sage: S.<x> = k['x', Frob]
+    sage: S.is_commutative()
+    False
+    sage: T.<y> = k['y', Frob^3]
+    sage: T.is_commutative()
+    True
+
+    sage: R.<t> = GF(5)[]
+    sage: der = R.derivation()
+    sage: A.<d> = R['d', der]
+    sage: A.is_commutative()
+    False
+    sage: B.<b> = R['b', 5*der]
+    sage: B.is_commutative()
+    True
+
 AUTHOR:
 
 - Xavier Caruso (2020-04)
@@ -21,25 +45,21 @@
 #                  https://www.gnu.org/licenses/
 # ***************************************************************************
 
-from sage.misc.prandom import randint
+from sage.categories.algebras import Algebras
+from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.morphism import Morphism
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_import import lazy_import
+from sage.misc.prandom import randint
 from sage.rings.infinity import Infinity
+from sage.rings.integer import Integer
+from sage.rings.polynomial.ore_polynomial_element import OrePolynomialBaseringInjection
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.rings.ring import _Fields
 from sage.structure.category_object import normalize_names
+from sage.structure.element import Element
 from sage.structure.parent import Parent
-
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.rings.integer import Integer
-from sage.structure.element import Element
-
-from sage.categories.commutative_rings import CommutativeRings
-from sage.categories.algebras import Algebras
-from sage.rings.ring import _Fields
-
-from sage.categories.morphism import Morphism
-
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.polynomial.ore_polynomial_element import OrePolynomialBaseringInjection
 
 lazy_import('sage.rings.derivation', 'RingDerivation')
 
@@ -421,7 +441,11 @@ def __init__(self, base_ring, morphism, derivation, name, sparse, category=None)
         self._morphism = morphism
         self._derivation = derivation
         self._fraction_field = None
-        category = Algebras(base_ring).or_subcategory(category)
+        if morphism is None and derivation is None:
+            cat = Algebras(base_ring).Commutative()
+        else:
+            cat = Algebras(base_ring)
+        category = cat.or_subcategory(category)
         Parent.__init__(self, base_ring, names=name,
                         normalize=True, category=category)
 
@@ -506,7 +530,7 @@ def build(check):
             pass
         if isinstance(a, str):
             try:
-                from sage.misc.parser import Parser, LookupNameMaker
+                from sage.misc.parser import LookupNameMaker, Parser
                 R = self.base_ring()
                 p = Parser(Integer, R, LookupNameMaker({self.variable_name(): self.gen()}, R))
                 return self(p.parse(a))
@@ -1095,36 +1119,6 @@ def random_irreducible(self, degree=2, monic=True, *args, **kwds):
             if irred.is_irreducible():
                 return irred
 
-    def is_commutative(self) -> bool:
-        r"""
-        Return ``True`` if this Ore polynomial ring is commutative.
-
-        This holds if the twisting morphism is the identity and the
-        twisting derivation vanishes.
-
-        EXAMPLES::
-
-            sage: # needs sage.rings.finite_rings
-            sage: k.<a> = GF(5^3)
-            sage: Frob = k.frobenius_endomorphism()
-            sage: S.<x> = k['x', Frob]
-            sage: S.is_commutative()
-            False
-            sage: T.<y> = k['y', Frob^3]
-            sage: T.is_commutative()
-            True
-
-            sage: R.<t> = GF(5)[]
-            sage: der = R.derivation()
-            sage: A.<d> = R['d', der]
-            sage: A.is_commutative()
-            False
-            sage: B.<b> = R['b', 5*der]
-            sage: B.is_commutative()
-            True
-        """
-        return self._morphism is None and self._derivation is None
-
     def is_field(self, proof=False) -> bool:
         r"""
         Return always ``False`` since Ore polynomial rings are never fields.