Indecomposable Module

Indecomposable Module

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.

Indecomposable is a weaker notion than simple module: simple means "no proper submodule", while indecomposable "not expressible as ".

A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple, which is a direct sum of simple modules.