Editing Indecomposable module (section)

In [[abstract algebra]], a [[module (mathematics)|module]] is '''indecomposable''' if it is non-zero and cannot be written as a [[direct sum of modules|direct sum]] of two non-zero [[submodule]]s.{{sfn|Jacobson|2009|p=111|ps=none}}{{sfn|Roman|2008|loc=p. 158 §6|ps=none}}

Indecomposable is a weaker notion than [[simple module]] (which is also sometimes called [[irreducibility (mathematics)|irreducible]] module):
simple means "no proper submodule" {{nowrap|''N'' < ''M''}},
while indecomposable "not expressible as {{nowrap|1=''N'' ⊕ ''P'' = ''M''}}".

A direct sum of indecomposables is called '''completely decomposable''';{{fact|date=December 2019}}<!-- this terminology does not appear in the reference (Jacobson) --> this is weaker than being [[semisimple module|semisimple]], which is a direct sum of [[simple module]]s.

A direct sum decomposition of a module into indecomposable modules is called an [[indecomposable decomposition]].