Algebra ideals - Category Framework

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# Algebra ideals

class sage.categories.algebra_ideals.AlgebraIdeals(A)

  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()

    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()

    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
        []