Bimodules - Category Framework

Skip to content
Back to top
View this page
Edit this page

# Bimodules

class sage.categories.bimodules.Bimodules(left_base, right_base, name=None)

  Bases: `CategoryWithParameters`

  The category of \((R,S)\)-bimodules.

  For \(R\) and \(S\) rings, a \((R,S)\)-bimodule \(X\) is a left \(R\)-module and right \(S\)-module such that the left and right actions commute: \(r*(x*s) = (r*x)*s\).

  EXAMPLES:

  Sage

      sage: Bimodules(QQ, ZZ)
      Category of bimodules over Rational Field on the left and Integer Ring on the right
      sage: Bimodules(QQ, ZZ).super_categories()
      [Category of left modules over Rational Field, Category of right modules over Integer Ring]

  Python

      >>> from sage.all import *
      >>> Bimodules(QQ, ZZ)
      Category of bimodules over Rational Field on the left and Integer Ring on the right
      >>> Bimodules(QQ, ZZ).super_categories()
      [Category of left modules over Rational Field, Category of right modules over Integer Ring]

  class ElementMethods

    Bases: `object`

  class ParentMethods

    Bases: `object`

  additional_structure()

    Return `None`.

    Indeed, the category of bimodules defines no additional structure: a left and right module morphism between two bimodules is a bimodule morphism.

    See also

    `Category.additional_structure()`

    Todo

    Should this category be a `CategoryWithAxiom`?

    EXAMPLES:

    Sage

        sage: Bimodules(QQ, ZZ).additional_structure()

    Python

        >>> from sage.all import *
        >>> Bimodules(QQ, ZZ).additional_structure()

  classmethod an_instance()

    Return an instance of this class.

    EXAMPLES:

    Sage

        sage: Bimodules.an_instance()                                               # needs sage.rings.real_mpfr
        Category of bimodules over Rational Field on the left and Real Field with 53 bits of precision on the right

    Python

        >>> from sage.all import *
        >>> Bimodules.an_instance()                                               # needs sage.rings.real_mpfr
        Category of bimodules over Rational Field on the left and Real Field with 53 bits of precision on the right

  left_base_ring()

    Return the left base ring over which elements of this category are defined.

    EXAMPLES:

    Sage

        sage: Bimodules(QQ, ZZ).left_base_ring()
        Rational Field

    Python

        >>> from sage.all import *
        >>> Bimodules(QQ, ZZ).left_base_ring()
        Rational Field

  right_base_ring()

    Return the right base ring over which elements of this category are defined.

    EXAMPLES:

    Sage

        sage: Bimodules(QQ, ZZ).right_base_ring()
        Integer Ring

    Python

        >>> from sage.all import *
        >>> Bimodules(QQ, ZZ).right_base_ring()
        Integer Ring

  super_categories()

    EXAMPLES:

    Sage

        sage: Bimodules(QQ, ZZ).super_categories()
        [Category of left modules over Rational Field, Category of right modules over Integer Ring]

    Python

        >>> from sage.all import *
        >>> Bimodules(QQ, ZZ).super_categories()
        [Category of left modules over Rational Field, Category of right modules over Integer Ring]