Bases: sage.categories.category_types.Category_module
The category of all modules over a base ring.
R-left and R-right modules modules over a commutative ring ## r*(x*s) = (r*x)*s
EXAMPLES:
sage: Modules(RationalField())
Category of modules over Rational Field
sage: Modules(Integers(9))
Category of modules over Ring of integers modulo 9
sage: Modules(Integers(9)).super_categories()
[Category of bimodules over Ring of integers modulo 9 on the left and Ring of integers modulo 9 on the right]
sage: Modules(Integers(9)).all_super_categories()
[Category of modules over Ring of integers modulo 9,
Category of bimodules over Ring of integers modulo 9 on the left and Ring of integers modulo 9 on the right,
Category of left modules over Ring of integers modulo 9,
Category of right modules over Ring of integers modulo 9,
Category of commutative additive groups,
Category of commutative additive monoids,
Category of commutative additive semigroups,
Category of additive magmas,
Category of sets,
Category of sets with partial maps,
Category of objects]
sage: Modules(ZZ).super_categories()
[Category of bimodules over Integer Ring on the left and Integer Ring on the right]
sage: Modules == RingModules
True
sage: Modules(ZZ[x]).is_abelian() # see #6081
True
TESTS:
sage: TestSuite(Modules(ZZ)).run()
TODO:
- When R is a field, Modules(R) could return VectorSpaces(R).
- Implement a FreeModules(R) category, when so prompted by a concrete use case
Bases: sage.categories.modules.Modules.HomCategory
The category of endomorphisms sets End(X) for X module (this is not used yet)
EXAMPLES:
sage: Hom(ZZ^3, ZZ^3).category().extra_super_categories() # todo: not implemented
[Category of algebras over Integer Ring]
Bases: sage.categories.category.HomCategory
The category of homomorphisms sets Hom(X,Y) for X, Y modules
EXAMPLES:
sage: Modules(ZZ).hom_category().extra_super_categories()
[Category of modules over Integer Ring]
EXAMPLES:
sage: Modules(QQ).super_categories()
[Category of bimodules over Rational Field on the left and Rational Field on the right]