A submodule is a subset of a module that is a subgroup of its additive group and is closed under multiplication by elements of the main ring . In particular, the left (right) ideal of the ring is a submodule of the left (right) -module .
Related Definitions
- A submodule that is different from the entire module is called its own .
- A submodule is called large (or significant ) if it has a nonzero intersection with any other nonzero submodule.
- For example, integers form a large submodule of a group of rational numbers.
- Each module is a large submodule of its injective shell .
- Submodule module is called small (or tangential ) if for any submodule equality entails .
- It turns out to be small, for example, every proper submodule of a chain module .
Properties
- The set of submodules of this module, ordered by inclusion, is a complete Dedekind lattice .
- The sum of all small submodules coincides with the intersection of all maximal submodules.
- Left ideal belongs to the Jacobson radical if and only if small in for every finitely generated left module .
- Elements of a small submodule are non-forming, that is, any system of generators of a module remains so after removing any of these elements (this, of course, does not mean that they can be deleted all at once!).
- The Jacobson radical of the module endomorphism ring coincides with the set of endomorphisms having a small image.
- If - module homomorphism to module then the set
turns out to be a submodule of the module and is called the kernel of homomorphism .- Each submodule serves as the kernel of a homomorphism.
Literature
- Kash F. Modules and rings, - per. with it., M. , 1981;
- Faith K. Algebra: rings, modules and categories, - per. with English., t. 1-2, M. , 1977-79.