Let $R$ be a ring. Recall that $R$ -module $M$ is called semisimple iff $M$ is a direct sum of simple module.

Proposition. The following are equivalent for $R$ -module $M$ :

1. $M$ is semisimple;
2. $M$ is generated by its simple submodules;
3. for every submodule $N\subseteq M$ there exists a submodule $N'\subseteq M$ such that $M=N\oplus N'$ .