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r""" Algebra modules """ 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_module from .modules import Modules
class AlgebraModules(Category_module): """ The category of modules over a fixed algebra $A$.
EXAMPLES::
sage: AlgebraModules(QQ['a']) Category of algebra modules over Univariate Polynomial Ring in a over Rational Field sage: AlgebraModules(QQ['a']).super_categories() [Category of modules over Univariate Polynomial Ring in a over Rational Field]
Note: as of now, `A` is required to be commutative, ensuring that the categories of left and right modules are isomorphic. Feedback and use cases for potential generalizations to the non commutative case are welcome.
""" def __init__(self, A): """ EXAMPLES::
sage: AlgebraModules(QQ['a']) Category of algebra modules over Univariate Polynomial Ring in a over Rational Field sage: AlgebraModules(QQ['a,b']) # todo: not implemented (QQ['a,b'] should be in Algebras(QQ)) sage: AlgebraModules(FreeAlgebra(QQ,2,'a,b')) Traceback (most recent call last): ... TypeError: A (=Free Algebra on 2 generators (a, b) over Rational Field) must be a commutative algebra sage: AlgebraModules(QQ) Traceback (most recent call last): ... TypeError: A (=Rational Field) must be a commutative algebra
TESTS::
sage: TestSuite(AlgebraModules(QQ['a'])).run() """
@classmethod def an_instance(cls): """ Returns an instance of this class
EXAMPLES::
sage: AlgebraModules.an_instance() Category of algebra modules over Univariate Polynomial Ring in x over Rational Field """
def algebra(self): """ EXAMPLES::
sage: AlgebraModules(QQ['x']).algebra() Univariate Polynomial Ring in x over Rational Field """
def super_categories(self): """ EXAMPLES::
sage: AlgebraModules(QQ['x']).super_categories() [Category of modules over Univariate Polynomial Ring in x over Rational Field] """ |