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. \(r.n \in N\) for all \(r \in R\)

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.