left R-module (the scalar multiplication is matrix multiplication). This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of (finitely or infinitely many) copies of this module Kn. Facts Every simple module is indecomposable. The converse is not true in general, as is shown by the second example above. By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an idempotent element different from 0 and 1. (If f is such an idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).) A module of finite length is indecomposable if and only if its endomorphism ring is local. Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma. In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull–Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic). Notes References Module theory