r"""
AlgebraIdeals
"""
from sage.misc.cachefunc import cached_method
from category_types import Category_ideal
from algebra_modules import AlgebraModules
from commutative_algebras import CommutativeAlgebras
class AlgebraIdeals(Category_ideal):
"""
The category of two-sided ideals in a fixed algebra `A`.
EXAMPLES::
sage: AlgebraIdeals(FreeAlgebra(QQ,2,'a,b'))
Category of algebra ideals in Free Algebra on 2 generators (a, b) over Rational Field
TODO:
- If useful, implement AlgebraLeftIdeals and AlgebraRightIdeals
of which AlgebraIdeals would be a subcategory
- Make AlgebraIdeals(R), return CommutativeAlgebraIdeals(R) when R is
commutative
"""
def __init__(self, A):
"""
EXAMPLES::
sage: AlgebraIdeals(FreeAlgebra(QQ,2,'a,b'))
Category of algebra ideals in Free Algebra on 2 generators (a, b) over Rational Field
sage: AlgebraIdeals(QQ)
Traceback (most recent call last):
...
TypeError: A (=Rational Field) must be an algebra
TESTS::
sage: TestSuite(AlgebraIdeals(FreeAlgebra(QQ,2,'a,b'))).run()
"""
from sage.algebras.algebra import is_Algebra
if not is_Algebra(A):
raise TypeError, "A (=%s) must be an algebra"%A
Category_ideal.__init__(self, A)
def algebra(self):
"""
EXAMPLES::
sage: AlgebraIdeals(QQ[x]).algebra()
Univariate Polynomial Ring in x over Rational Field
"""
return self.ambient()
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()
[]
"""
R = self.algebra()
try:
if R.is_commutative():
return [AlgebraModules(R)]
except (AttributeError,NotImplementedError):
pass
return []
