submodule

A submodule is a piece of a module that is itself a module under the same operations — the module-theoretic analogue of a subspace of a vector space or a subgroup of a group. It is the part you are allowed to restrict to: a subset that stays closed when you add elements together and when you scale them.

Formally, a subset N of a left R-module M is a submodule if N is a subgroup of (M, +) and rn lies in N for every r in R and n in N. Equivalently (the submodule criterion), N is a nonempty subset with rn + n' in N whenever r is in R and n, n' are in N. The submodules of a module form a lattice under intersection and sum, and this lattice often encodes deep structure.

The submodules of R viewed as a module over itself are exactly its left ideals — so ideal theory is a special case of submodule theory. Sums N + N' = {n + n' : n in N, n' in N'} and intersections of submodules are again submodules, but unlike subspaces, two submodules generally have no direct complement; that failure is precisely what makes projective and injective modules interesting.