[[Module theory MOC]]
# Reducibility of modules

Let $M$ be a (left) $R$-[[module]].
We say that $M$ is

- **[[Simple module|simple]]** or **irreducible** iff it has no nonzero proper [[submodule]];
- **[[Semisimple module|semisimple]]** or **completely reducible** iff it is the [[Direct sum of modules|direct sum]] of simple modules;
- **[[Decomposable module|decomposable]]** iff it is the direct sum of two nonzero modules;
- **[[Indecomposable module|indecomposable]]** iff it is not decomposable.

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