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r""" AlgebraIdeals """ from __future__ import absolute_import #***************************************************************************** # Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu> # William Stein <wstein@math.ucsd.edu> # 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from .category_types import Category_ideal from .algebra_modules import AlgebraModules
class AlgebraIdeals(Category_ideal): """ The category of two-sided ideals in a fixed algebra `A`.
EXAMPLES::
sage: AlgebraIdeals(QQ['a']) Category of algebra ideals in Univariate Polynomial Ring in a over Rational Field
.. TODO::
- Add support for non commutative rings (this is currently not supported by the subcategory :class:`AlgebraModules`). - Make ``AlgebraIdeals(R)``, return ``CommutativeAlgebraIdeals(R)`` when ``R`` is commutative. - If useful, implement ``AlgebraLeftIdeals`` and ``AlgebraRightIdeals`` of which ``AlgebraIdeals`` would be a subcategory. """ def __init__(self, A): """ EXAMPLES::
sage: AlgebraIdeals(QQ['a']) Category of algebra ideals in Univariate Polynomial Ring in a over Rational Field sage: AlgebraIdeals(QQ) Traceback (most recent call last): ... TypeError: A (=Rational Field) must be an algebra
TESTS::
sage: TestSuite(AlgebraIdeals(QQ['a'])).run() """
def algebra(self): """ EXAMPLES::
sage: AlgebraIdeals(QQ['x']).algebra() Univariate Polynomial Ring in x over Rational Field """
def super_categories(self): """ The category of algebra modules should be a super category of this category.
However, since algebra modules are currently only available over commutative rings, we have to omit it if our ring is non-commutative.
EXAMPLES::
sage: AlgebraIdeals(QQ['x']).super_categories() [Category of algebra modules over Univariate Polynomial Ring in x over Rational Field] sage: C = AlgebraIdeals(FreeAlgebra(QQ,2,'a,b')) sage: C.super_categories() []
""" except (AttributeError,NotImplementedError): pass |