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.