Indecomposable module

Summary

An indecomposable module is a nonzero module that cannot be written as a direct sum of two nonzero submodules. Every simple module is indecomposable, but an indecomposable module may have many submodules. By the Krull–Schmidt theorem, a module of finite length is a direct sum of indecomposable modules in an essentially unique way. So classifying modules of finite length over a ring comes down to classifying its indecomposable modules.