using Parent in ring extensions

src/sage/rings/morphism.pyx

@@ -2879,8 +2879,10 @@ cdef class FrobeniusEndomorphism_generic(RingHomomorphism):
              over Finite Field of size 5
         """
         from sage.rings.ring import CommutativeRing
+        from sage.categories.commutative_rings import CommutativeRings
         from sage.categories.homset import Hom
-        if not isinstance(domain, CommutativeRing):
+        if not (domain in CommutativeRings() or
+                isinstance(domain, CommutativeRing)):  # TODO: remove this line
             raise TypeError("The base ring must be a commutative ring")
         self._p = domain.characteristic()
         if not self._p.is_prime():

src/sage/rings/number_field/order.py

@@ -77,8 +77,9 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+from sage.categories.integral_domains import IntegralDomains
 from sage.misc.cachefunc import cached_method
-from sage.rings.ring import IntegralDomain
+from sage.structure.parent import Parent
 from sage.structure.sequence import Sequence
 from sage.rings.integer_ring import ZZ
 import sage.rings.abc
@@ -426,7 +427,7 @@ def EquationOrder(f, names, **kwds):
     return K.order(K.gens())
 
 
-class Order(IntegralDomain, sage.rings.abc.Order):
+class Order(Parent, sage.rings.abc.Order):
     r"""
     An order in a number field.
 
@@ -478,8 +479,8 @@ def __init__(self, K):
             0.0535229072603327 + 1.20934552493846*I
         """
         self._K = K
-        IntegralDomain.__init__(self, ZZ, names=K.variable_names(),
-                                normalize=False)
+        Parent.__init__(self, base=ZZ, names=K.variable_names(),
+                        normalize=False, category=IntegralDomains())
         self._populate_coercion_lists_(embedding=self.number_field())
         if self.absolute_degree() == 2:
             self.is_maximal()       # cache
@@ -665,7 +666,7 @@ def krull_dimension(self):
             sage: O2.krull_dimension()
             1
         """
-        return ZZ(1)
+        return ZZ.one()
 
     def integral_closure(self):
         r"""
@@ -733,6 +734,20 @@ def ngens(self):
         """
         return self.absolute_degree()
 
+    def gens(self) -> tuple:
+        """
+        Return the generators as a tuple.
+
+        EXAMPLES::
+
+            sage: x = polygen(ZZ, 'x')
+            sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
+            sage: O = K.maximal_order()
+            sage: O.gens()
+            (1, 1/2*a^2 + 1/2*a, a^2)
+        """
+        return tuple(self.gen(i) for i in range(self.absolute_degree()))
+
     def basis(self):  # this must be defined in derived class
         r"""
         Return a basis over `\ZZ` of this order.

src/sage/rings/ring_extension.pxd

@@ -1,8 +1,8 @@
 from sage.categories.map cimport Map
-from sage.rings.ring cimport CommutativeRing
+from sage.structure.parent cimport Parent
 
 
-cdef class RingExtension_generic(CommutativeRing):
+cdef class RingExtension_generic(Parent):
     cdef _type
     cdef _backend
     cdef _defining_morphism
@@ -15,10 +15,10 @@ cdef class RingExtension_generic(CommutativeRing):
     cdef type _fraction_field_type
 
     cpdef is_defined_over(self, base)
-    cpdef CommutativeRing _check_base(self, CommutativeRing base)
-    cpdef _degree_over(self, CommutativeRing base)
-    cpdef _is_finite_over(self, CommutativeRing base)
-    cpdef _is_free_over(self, CommutativeRing base)
+    cpdef Parent _check_base(self, Parent base)
+    cpdef _degree_over(self, Parent base)
+    cpdef _is_finite_over(self, Parent base)
+    cpdef _is_free_over(self, Parent base)
     cdef Map _defining_morphism_fraction_field(self, bint extend_base)
 
 
@@ -31,10 +31,11 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
     cdef _basis_names
     cdef _basis_latex_names
 
-    cpdef _basis_over(self, CommutativeRing base)
-    # cpdef _free_module(self, CommutativeRing base, bint map)
+    cpdef _basis_over(self, Parent base)
+    # cpdef _free_module(self, Parent base, bint map)
 
 
 cdef class RingExtensionWithGen(RingExtensionWithBasis):
     cdef _gen
     cdef _name
+    cdef public object _latex_names

src/sage/rings/ring_extension.pyx

@@ -117,12 +117,12 @@ from sage.cpython.getattr import dir_with_other_class
 from sage.misc.latex import latex, latex_variable_name
 
 from sage.structure.factory import UniqueFactory
+from sage.structure.parent cimport Parent
 from sage.structure.element cimport Element
 from sage.structure.category_object import normalize_names
 from sage.categories.map cimport Map
 from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.fields import Fields
-from sage.rings.ring cimport CommutativeRing
 from sage.rings.integer_ring import ZZ
 from sage.rings.infinity import Infinity
 
@@ -497,7 +497,7 @@ RingExtension = RingExtensionFactory("sage.rings.ring_extension.RingExtension")
 # General extensions
 ####################
 
-cdef class RingExtension_generic(CommutativeRing):
+cdef class RingExtension_generic(Parent):
     r"""
     A generic class for all ring extensions.
 
@@ -564,8 +564,8 @@ cdef class RingExtension_generic(CommutativeRing):
             ...
             ValueError: exotic defining morphism between two rings in the tower; consider using another variable name
         """
-        cdef CommutativeRing base, ring
-        cdef CommutativeRing b, backend
+        cdef Parent base, ring
+        cdef Parent b, backend
         cdef Map f
 
         base = defining_morphism.domain()
@@ -575,7 +575,7 @@ cdef class RingExtension_generic(CommutativeRing):
             # but CommutativeRings() seems safer, especially when dealing with
             # morphisms which do not need to preserve the base
             category = CommutativeRings()
-        CommutativeRing.__init__(self, ZZ, category=category)
+        Parent.__init__(self, ZZ, category=category)
         self._base = base
         self._backend = ring
         self._backend_defining_morphism = defining_morphism
@@ -1197,14 +1197,14 @@ cdef class RingExtension_generic(CommutativeRing):
 
             :meth:`base`, :meth:`bases`, :meth:`absolute_base`
         """
-        cdef CommutativeRing b
+        cdef Parent b
         b = self
         while isinstance(b, RingExtension_generic):
             if b is base or (<RingExtension_generic>b)._backend is base: return True
             b = (<RingExtension_generic>b)._base
         return b is base
 
-    cpdef CommutativeRing _check_base(self, CommutativeRing base):
+    cpdef Parent _check_base(self, Parent base):
         r"""
         Check if ``base`` is one of the successive bases of this
         extension and, if it is, normalize it.
@@ -1243,7 +1243,7 @@ cdef class RingExtension_generic(CommutativeRing):
             sage: L._check_base(None) is L.base()                                       # needs sage.rings.finite_rings
             True
         """
-        cdef CommutativeRing b
+        cdef Parent b
         if base is None:
             return self._base
         b = self
@@ -1450,7 +1450,7 @@ cdef class RingExtension_generic(CommutativeRing):
         base = self._check_base(base)
         return self._degree_over(base)
 
-    cpdef _degree_over(self, CommutativeRing base):
+    cpdef _degree_over(self, Parent base):
         r"""
         Return the degree of this extension over ``base``.
 
@@ -1572,7 +1572,7 @@ cdef class RingExtension_generic(CommutativeRing):
             sage: L.is_finite_over()                                                    # needs sage.rings.finite_rings
             True
         """
-        cdef CommutativeRing b
+        cdef Parent b
         base = self._check_base(base)
         if base is self:
             return True
@@ -1590,7 +1590,7 @@ cdef class RingExtension_generic(CommutativeRing):
             b = (<RingExtension_generic?>b)._base
         raise NotImplementedError
 
-    cpdef _is_finite_over(self, CommutativeRing base):
+    cpdef _is_finite_over(self, Parent base):
         r"""
         Return whether or not this extension is finite over ``base``.
 
@@ -1635,7 +1635,7 @@ cdef class RingExtension_generic(CommutativeRing):
             sage: L.is_free_over()                                                      # needs sage.rings.finite_rings
             True
         """
-        cdef CommutativeRing b
+        cdef Parent b
         base = self._check_base(base)
         if base is self or base.is_field():
             return True
@@ -1653,7 +1653,7 @@ cdef class RingExtension_generic(CommutativeRing):
             b = (<RingExtension_generic?>b)._base
         raise NotImplementedError
 
-    cpdef _is_free_over(self, CommutativeRing base):
+    cpdef _is_free_over(self, Parent base):
         r"""
         Return whether or not this extension is finite over ``base``.
 
@@ -2191,7 +2191,7 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
             b = b.base_ring()
         return base
 
-    cpdef _degree_over(self, CommutativeRing base):
+    cpdef _degree_over(self, Parent base):
         r"""
         Return the degree of this extension over ``base``.
 
@@ -2218,7 +2218,7 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
         else:
             return len(self._basis) * self._base._degree_over(base)
 
-    cpdef _is_finite_over(self, CommutativeRing base):
+    cpdef _is_finite_over(self, Parent base):
         r"""
         Return whether or not this extension is finite over ``base``.
 
@@ -2237,7 +2237,7 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
             return True
         return self._base._is_finite_over(base)
 
-    cpdef _is_free_over(self, CommutativeRing base):
+    cpdef _is_free_over(self, Parent base):
         r"""
         Return whether or not this extension is free over ``base``.
 
@@ -2298,7 +2298,7 @@ cdef class RingExtensionWithBasis(RingExtension_generic):
         base = self._check_base(base)
         return self._basis_over(base)
 
-    cpdef _basis_over(self, CommutativeRing base):
+    cpdef _basis_over(self, Parent base):
         r"""
         Return a basis of this extension over ``base``.
 
@@ -2588,7 +2588,6 @@ cdef class RingExtensionWithGen(RingExtensionWithBasis):
         RingExtensionWithBasis.__init__(self, defining_morphism, basis, basis_names, check, **kwargs)
         self._gen = self._backend(gen)
         self._names = (self._name,)
-        self._latex_names = (latex_variable_name(self._name),)
         self._basis_latex_names = basis_latex_names
 
     def _repr_topring(self, **options):

src/sage/rings/ring_extension_element.pxd

@@ -1,4 +1,4 @@
-from sage.rings.ring cimport CommutativeRing
+from sage.structure.parent cimport Parent
 from sage.structure.element cimport Element
 from sage.structure.element cimport CommutativeAlgebraElement
 from sage.rings.ring_extension cimport RingExtension_generic
@@ -13,10 +13,10 @@ cdef class RingExtensionFractionFieldElement(RingExtensionElement):
     pass
 
 cdef class RingExtensionWithBasisElement(RingExtensionElement):
-    cdef _vector(self, CommutativeRing base)
-    cdef _matrix(self, CommutativeRing base)
-    cdef _trace(self, CommutativeRing base)
-    cdef _norm(self, CommutativeRing base)
+    cdef _vector(self, Parent base)
+    cdef _matrix(self, Parent base)
+    cdef _trace(self, Parent base)
+    cdef _norm(self, Parent base)
     cpdef minpoly(self, base=*, var=*)
 
 

src/sage/rings/ring_extension_element.pyx

@@ -24,6 +24,7 @@ from sage.misc.latex import latex
 
 from sage.structure.category_object import normalize_names
 from sage.structure.element cimport CommutativeAlgebraElement
+from sage.structure.parent cimport Parent
 from sage.rings.integer_ring import ZZ
 from sage.categories.fields import Fields
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -1033,7 +1034,7 @@ cdef class RingExtensionWithBasisElement(RingExtensionElement):
         base = (<RingExtension_generic>self._parent)._check_base(base)
         return self._vector(base)
 
-    cdef _vector(self, CommutativeRing base):
+    cdef _vector(self, Parent base):
         r"""
         Return the vector of coordinates of this element over ``base``
         (in the basis output by the method :meth:`basis_over`).
@@ -1198,7 +1199,7 @@ cdef class RingExtensionWithBasisElement(RingExtensionElement):
             raise ValueError("the extension is not finite free")
         return self._matrix(base)
 
-    cdef _matrix(self, CommutativeRing base):
+    cdef _matrix(self, Parent base):
         r"""
         Return the matrix of the multiplication by this element (in
         the basis output by :meth:`basis_over`).
@@ -1286,7 +1287,7 @@ cdef class RingExtensionWithBasisElement(RingExtensionElement):
             raise ValueError("the extension is not finite free")
         return self._trace(base)
 
-    cdef _trace(self, CommutativeRing base):
+    cdef _trace(self, Parent base):
         r"""
         Return the trace of this element over ``base``.
 
@@ -1314,7 +1315,7 @@ cdef class RingExtensionWithBasisElement(RingExtensionElement):
             ....:     assert((x+y).trace(base) == x.trace(base) + y.trace(base))
         """
         cdef RingExtensionWithBasis parent = self._parent
-        cdef CommutativeRing b
+        cdef Parent b
         if base is parent:
             return self
         b = parent._base
@@ -1379,7 +1380,7 @@ cdef class RingExtensionWithBasisElement(RingExtensionElement):
             raise ValueError("the extension is not finite free")
         return self._norm(base)
 
-    cdef _norm(self, CommutativeRing base):
+    cdef _norm(self, Parent base):
         r"""
         Return the norm of this element over ``base``.
 
@@ -1407,7 +1408,7 @@ cdef class RingExtensionWithBasisElement(RingExtensionElement):
             ....:     assert((x*y).norm(base) == x.norm(base) * y.norm(base))
         """
         cdef RingExtensionWithBasis parent = self._parent
-        cdef CommutativeRing b
+        cdef Parent b
         if base is parent:
             return self
         b = parent._base

src/sage/rings/ring_extension_morphism.pyx

@@ -21,7 +21,7 @@ from sage.structure.richcmp import op_EQ, op_NE
 
 from sage.structure.element cimport Element
 from sage.categories.map import Map
-from sage.rings.ring cimport CommutativeRing
+from sage.structure.parent cimport Parent
 from sage.rings.morphism cimport RingMap
 from sage.rings.ring_extension cimport RingExtension_generic, RingExtensionWithBasis
 from sage.rings.ring_extension_element cimport RingExtensionElement
@@ -57,7 +57,7 @@ cdef are_equal_morphisms(f, g):
         sage: H(f^2) == H(g^2)  # indirect doctest
         True
     """
-    cdef CommutativeRing b
+    cdef Parent b
     cdef tuple gens
     if f is None and g is None:
         return True
@@ -821,7 +821,7 @@ cdef class MapRelativeRingToFreeModule(Map):
               From: Field in z6 with defining polynomial x^2 + (10*z3^2 + z3 + 6)*x + z3 over its base
               To:   Vector space of dimension 2 over Finite Field in z3 of size 11^3
         """
-        cdef CommutativeRing L, base
+        cdef Parent L, base
 
         self._degree = (<RingExtensionWithBasis>E)._degree_over(K)
         self._basis = [ (<RingExtensionElement>x)._backend for x in E.basis_over(K) ]