In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.[1][2]
Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module):
simple means "no proper submodule" N < M,
while indecomposable "not expressible as N ⊕ P = M".
A direct sum of indecomposables is called completely decomposable;[citation needed] this is weaker than being semisimple, which is a direct sum of simple modules.
A direct sum decomposition of a module into indecomposable modules is called an indecomposable decomposition.
^Jacobson 2009, p. 111
^Roman 2008, p. 158 §6
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