Algebra ideals#

class sage.categories.algebra_ideals.AlgebraIdeals(A)[source]#

Bases: Category_ideal

The category of two-sided ideals in a fixed algebra \(A\).

EXAMPLES:

Sage

sage: AlgebraIdeals(QQ['a'])
Category of algebra ideals in Univariate Polynomial Ring in a over Rational Field

Python

>>> from sage.all import *
>>> 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 AlgebraModules).
Make AlgebraIdeals(R), return CommutativeAlgebraIdeals(R) when R is commutative.
If useful, implement AlgebraLeftIdeals and AlgebraRightIdeals of which AlgebraIdeals would be a subcategory.

algebra()[source]#

EXAMPLES:

Sage

sage: AlgebraIdeals(QQ['x']).algebra()
Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> AlgebraIdeals(QQ['x']).algebra()
Univariate Polynomial Ring in x over Rational Field

super_categories()[source]#

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

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'))                          # needs sage.combinat sage.modules
sage: C.super_categories()                                                  # needs sage.combinat sage.modules
[]

Python

>>> from sage.all import *
>>> AlgebraIdeals(QQ['x']).super_categories()
[Category of algebra modules
  over Univariate Polynomial Ring in x over Rational Field]
>>> C = AlgebraIdeals(FreeAlgebra(QQ, Integer(2), 'a,b'))                          # needs sage.combinat sage.modules
>>> C.super_categories()                                                  # needs sage.combinat sage.modules
[]