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	2013-03-22 18:53:27
Last modified on	2013-03-22 18:53:27
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Theorem
Classification	msc 16D60