Semisimple modules and short exact sequences Though I suspect that this is a folklore result, I've been unable to find a proof by digital book searching. I also mention (my apologies) that I am new to noncommutative algebras theory and I am not familiar with the literature.


  Let $0 \to M \to N \to P \to 0$ be a short exact sequence of left $A$-modules ($A$ some arbitrary ring). If $M$ and $P$ are semisimple, prove that $N$ is semisimple.


Closest I could find is the (well-known) fact that submodules/quotients of semisimple modules are semisimple.

I would be grateful for any indication or relevant reference.
The sequence $$0\to R/(I \cap J) \to R/I \oplus R/J \to R/(I+J) \to 0    $$ 
is exact. Since $R/I$ and $R/J$ are semisimple, $R/I \oplus R/J $ is semisimple, and then $R/(I+J)$ is semisimple.
The sequence $$0\to R/(I \cap J) \to R/I \oplus R/J \to R/(I+J) \to 0    $$ 
is exact. Since $R/I$ and $R/J$ are semisimple, $R/I \oplus R/J $ is semisimple, and then $R/(I+J)$ is semisimple.