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: 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]

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: Bimodules(QQ, ZZ).additional_structure()

classmethod an_instance()#

Return an instance of this class.

EXAMPLES:

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

super_categories()#

EXAMPLES:

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