Submodule

Theorem

Given a ring R and a R-module M, a subset N of M is called a submodule if

  1. N is a subgroup of M
  2. r.nN for all rR

This sub-object of a module is natural in that it imposes the usual closure and structure preserving operations.


Taking M as a G-set with the group G=R, then submodules are exactly the G-stable subsets which are also subgroups of M.

In the case of a ring R being treated as an R-module, the submodules of R are exactly the ideals of R.

If the module is a vector space, that is, R is a field, then submodules correspond with vector subspace.