Semisimple iff every submodule is a direct summand

Semisimple iff every submodule is a direct summand: For an $R$-module $M$, the following are equivalent:

1. $M$ is semisimple.
2. Every submodule $N\le M$ is a direct summand of $M$, i.e. there exists a submodule $N'\le M$ with $M = N\oplus N'$.
3. $M$ is (isomorphic to) a direct sum of simple submodules.

This gives the splitting characterization of semisimple modules in terms of submodules , and explains why such modules decompose as direct sums of simple modules .