moved some is_noetherian to categories

Author: fchapoton

src/sage/algebras/steenrod/steenrod_algebra.py

@@ -3059,7 +3059,7 @@ def is_integral_domain(self, proof=True):
         """
         return self.is_field()
 
-    def is_noetherian(self):
+    def is_noetherian(self) -> bool:
         """
         This algebra is Noetherian if and only if it is finite.
 

src/sage/rings/finite_rings/integer_mod_ring.py

@@ -249,10 +249,10 @@ def create_object(self, version, order, **kwds):
 Zmod = Integers = IntegerModRing = IntegerModFactory("IntegerModRing")
 
 
-from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.noetherian_rings import NoetherianRings
 from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
 from sage.categories.category import JoinCategory
-default_category = JoinCategory((CommutativeRings(), FiniteEnumeratedSets()))
+default_category = JoinCategory((NoetherianRings(), FiniteEnumeratedSets()))
 ZZ = integer_ring.IntegerRing()
 
 
@@ -448,6 +448,11 @@ def __init__(self, order, cache=None, category=None):
             sage: R = IntegerModRing(18)
             sage: R.is_finite()
             True
+
+        TESTS::
+
+            sage: Integers(8).is_noetherian()
+            True
         """
         order = ZZ(order)
         if order <= 0:
@@ -478,7 +483,7 @@ def __init__(self, order, cache=None, category=None):
         self._zero_element = integer_mod.IntegerMod(self, 0)
         self._one_element = integer_mod.IntegerMod(self, 1)
 
-    def _macaulay2_init_(self, macaulay2=None):
+    def _macaulay2_init_(self, macaulay2=None) -> str:
         """
         EXAMPLES::
 
@@ -498,7 +503,7 @@ def _macaulay2_init_(self, macaulay2=None):
         """
         return "ZZ/{}".format(self.order())
 
-    def _axiom_init_(self):
+    def _axiom_init_(self) -> str:
         """
         Return a string representation of ``self`` in (Pan)Axiom.
 
@@ -529,17 +534,6 @@ def krull_dimension(self):
         """
         return integer.Integer(0)
 
-    def is_noetherian(self):
-        """
-        Check if ``self`` is a Noetherian ring.
-
-        EXAMPLES::
-
-            sage: Integers(8).is_noetherian()
-            True
-        """
-        return True
-
     def extension(self, poly, name=None, names=None, **kwds):
         """
         Return an algebraic extension of ``self``. See
@@ -560,7 +554,7 @@ def extension(self, poly, name=None, names=None, **kwds):
         return CommutativeRing.extension(self, poly, name, names, **kwds)
 
     @cached_method
-    def is_prime_field(self):
+    def is_prime_field(self) -> bool:
         """
         Return ``True`` if the order is prime.
 
@@ -573,7 +567,7 @@ def is_prime_field(self):
         """
         return self.__order.is_prime()
 
-    def _precompute_table(self):
+    def _precompute_table(self) -> None:
         """
         Compute a table of elements so that elements are unique.
 
@@ -585,7 +579,7 @@ def _precompute_table(self):
         """
         self._pyx_order.precompute_table(self)
 
-    def list_of_elements_of_multiplicative_group(self):
+    def list_of_elements_of_multiplicative_group(self) -> list:
         """
         Return a list of all invertible elements, as python ints.
 

src/sage/rings/number_field/order.py

@@ -78,6 +78,7 @@
 # ****************************************************************************
 
 from sage.categories.integral_domains import IntegralDomains
+from sage.categories.noetherian_rings import NoetherianRings
 from sage.misc.cachefunc import cached_method
 from sage.structure.parent import Parent
 from sage.structure.sequence import Sequence
@@ -457,6 +458,16 @@ class Order(Parent, sage.rings.abc.Order):
         Traceback (most recent call last):
         ...
         ValueError: the rank of the span of gens is wrong
+
+    Orders are always Noetherian::
+
+        sage: x = polygen(ZZ, 'x')
+        sage: L.<alpha> = NumberField(x**4 - x**2 + 7)
+        sage: O = L.maximal_order() ; O.is_noetherian()
+        True
+        sage: E.<w> = NumberField(x^2 - x + 2)
+        sage: OE = E.ring_of_integers(); OE.is_noetherian()
+        True
     """
 
     def __init__(self, K):
@@ -479,8 +490,9 @@ def __init__(self, K):
             0.0535229072603327 + 1.20934552493846*I
         """
         self._K = K
+        cat = IntegralDomains() & NoetherianRings()
         Parent.__init__(self, base=ZZ, names=K.variable_names(),
-                        normalize=False, category=IntegralDomains())
+                        normalize=False, category=cat)
         self._populate_coercion_lists_(embedding=self.number_field())
         if self.absolute_degree() == 2:
             self.is_maximal()       # cache
@@ -615,22 +627,6 @@ def is_field(self, proof=True):
         """
         return False
 
-    def is_noetherian(self):
-        r"""
-        Return ``True`` (because orders are always Noetherian).
-
-        EXAMPLES::
-
-            sage: x = polygen(ZZ, 'x')
-            sage: L.<alpha> = NumberField(x**4 - x**2 + 7)
-            sage: O = L.maximal_order() ; O.is_noetherian()
-            True
-            sage: E.<w> = NumberField(x^2 - x + 2)
-            sage: OE = E.ring_of_integers(); OE.is_noetherian()
-            True
-        """
-        return True
-
     def is_integrally_closed(self) -> bool:
         r"""
         Return whether this ring is integrally closed.
@@ -822,7 +818,7 @@ def coordinates(self, x):
             from sage.matrix.constructor import Matrix
             self.__basis_matrix_inverse = Matrix([to_V(b) for b in self.basis()]).inverse()
             M = self.__basis_matrix_inverse
-        return to_V(K(x))*M
+        return to_V(K(x)) * M
 
     def free_module(self):
         r"""
@@ -1375,7 +1371,8 @@ def random_element(self, *args, **kwds):
             sage: A.random_element().parent() is A
             True
         """
-        return sum([ZZ.random_element(*args, **kwds)*a for a in self.basis()])
+        return sum([ZZ.random_element(*args, **kwds) * a
+                    for a in self.basis()])
 
     def absolute_degree(self):
         r"""
@@ -1492,41 +1489,41 @@ def some_elements(self):
                 elements.append(self(a))
         return elements
 
-##     def absolute_polynomial(self):
-##         """
-##         Return the absolute polynomial of this order, which is just the absolute polynomial of the number field.
+#     def absolute_polynomial(self):
+#         """
+#         Return the absolute polynomial of this order, which is just the absolute polynomial of the number field.
 
-##         EXAMPLES::
+#         EXAMPLES::
 
-##         sage: K.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); OK = K.maximal_order()
-##         Traceback (most recent call last):
-##         ...
-##         NotImplementedError
+#         sage: K.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); OK = K.maximal_order()
+#         Traceback (most recent call last):
+#         ...
+#         NotImplementedError
 
-##         #sage: OK.absolute_polynomial()
-##         #x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1
-##         """
-##         return self.number_field().absolute_polynomial()
+#         #sage: OK.absolute_polynomial()
+#         #x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1
+#         """
+#         return self.number_field().absolute_polynomial()
 
-##     def polynomial(self):
-##         """
-##         Return the polynomial defining the number field that contains self.
-##         """
-##         return self.number_field().polynomial()
+#     def polynomial(self):
+#         """
+#         Return the polynomial defining the number field that contains self.
+#         """
+#         return self.number_field().polynomial()
 
-##     def polynomial_ntl(self):
-##         """
-##         Return defining polynomial of the parent number field as a
-##         pair, an ntl polynomial and a denominator.
+#     def polynomial_ntl(self):
+#         """
+#         Return defining polynomial of the parent number field as a
+#         pair, an ntl polynomial and a denominator.
 
-##         This is used mainly to implement some internal arithmetic.
+#         This is used mainly to implement some internal arithmetic.
 
-##         EXAMPLES::
+#         EXAMPLES::
 
-##             sage: NumberField(x^2 + 1,'a').maximal_order().polynomial_ntl()
-##             ([1 0 1], 1)
-##         """
-##         return self.number_field().polynomial_ntl()
+#             sage: NumberField(x^2 + 1,'a').maximal_order().polynomial_ntl()
+#             ([1 0 1], 1)
+#         """
+#         return self.number_field().polynomial_ntl()
 
 
 class Order_absolute(Order):
@@ -1603,7 +1600,7 @@ def _element_constructor_(self, x):
             3*a^2 + 2*a + 1
         """
         if isinstance(x, (tuple, list)):
-            x = sum(xi*gi for xi, gi in zip(x, self.gens()))
+            x = sum(xi * gi for xi, gi in zip(x, self.gens()))
         if not isinstance(x, Element) or x.parent() is not self._K:
             x = self._K(x)
         V, _, embedding = self._K.vector_space()