Bases: sage.categories.category_types.Category_module
The category of all modules over a base ring
A -module
is a left and right
-module over a commutative
ring
such that:
INPUT:
- base_ring – a ring
- dispatch – a boolean (for internal use; default: True)
When the base ring is a field, the category of vector spaces is returned instead (unless dispatch == False).
EXAMPLES:
sage: Modules(IntegerRing())
Category of modules over Integer Ring
sage: Modules(RationalField())
Category of vector spaces 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(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:
- Implement a FreeModules(R) category, when so prompted by a concrete use case
Bases: sage.categories.modules.Modules.HomCategory
The category of endomorphisms sets for
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 for
,
modules
alias of HomCategory.ParentMethods
EXAMPLES:
sage: Modules(ZZ).hom_category().extra_super_categories()
[Category of modules over Integer Ring]
EXAMPLES:
sage: Modules(ZZ).super_categories()
[Category of bimodules over Integer Ring on the left and Integer Ring on the right]
Nota bene:
sage: Modules(QQ)
Category of vector spaces over Rational Field
sage: Modules(QQ).super_categories()
[Category of modules over Rational Field]