properties of semisimple modules
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 ${N}^{\prime}\subseteq M$ such that $M=N\oplus {N}^{\prime}$.
Title  properties of semisimple modules 

Canonical name  PropertiesOfSemisimpleModules 
Date of creation  20130322 18:53:27 
Last modified on  20130322 18:53:27 
Owner  joking (16130) 
Last modified by  joking (16130) 
Numerical id  4 
Author  joking (16130) 
Entry type  Theorem 
Classification  msc 16D60 