Semisimple modules and short exact sequences (2)

Question

Let $R$ be a ring and $$0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$$ a short exact sequence. If $M$ is semisimple what can we say about $L$ and $N$?

I guess they are also semisimple. Is this true? What is the idea of the proof?

Answer 1

If $M$ is semisimple, then any submodule $L$ and the corresponding quotient module $N=M/L$ are semisimple. The proof is based on the fact that a semisimple module is the direct sum of simple submodules - see here.