Bimodules¶
- class sage.categories.bimodules.Bimodules(left_base, right_base, name=None)[source]¶
- 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] - >>> 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] - additional_structure()[source]¶
- 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() - >>> from sage.all import * >>> Bimodules(QQ, ZZ).additional_structure() 
 - classmethod an_instance()[source]¶
- Return an instance of this class. - EXAMPLES: - 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 - >>> 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()[source]¶
- Return the left base ring over which elements of this category are defined. - EXAMPLES: - sage: Bimodules(QQ, ZZ).left_base_ring() Rational Field - >>> from sage.all import * >>> Bimodules(QQ, ZZ).left_base_ring() Rational Field 
 - right_base_ring()[source]¶
- Return the right base ring over which elements of this category are defined. - EXAMPLES: - sage: Bimodules(QQ, ZZ).right_base_ring() Integer Ring - >>> from sage.all import * >>> Bimodules(QQ, ZZ).right_base_ring() Integer Ring 
 - super_categories()[source]¶
- EXAMPLES: - sage: Bimodules(QQ, ZZ).super_categories() [Category of left modules over Rational Field, Category of right modules over Integer Ring] - >>> from sage.all import * >>> Bimodules(QQ, ZZ).super_categories() [Category of left modules over Rational Field, Category of right modules over Integer Ring]