Bimodules¶
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class
sage.categories.bimodules.
Bimodules
(left_base, right_base, name=None)¶ Bases:
sage.categories.category.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]
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class
ElementMethods
¶ Bases:
object
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class
ParentMethods
¶ Bases:
object
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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
Todo
Should this category be a
CategoryWithAxiom
?EXAMPLES:
sage: Bimodules(QQ, ZZ).additional_structure()
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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
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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
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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
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super_categories
()¶ EXAMPLES:
sage: Bimodules(QQ, ZZ).super_categories() [Category of left modules over Rational Field, Category of right modules over Integer Ring]
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class