Editing Indecomposable module (section)

==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 (ring theory)#Role in decompositions|idempotent element]] different from 0 and 1.{{sfn|Jacobson|2009|p=111|ps=none}} (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 of a module|length]] is indecomposable if and only if its endomorphism ring is [[local ring|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 indecomposables, 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).{{sfn|Jacobson|2009|p=115|ps=none}}