# A question on isomorphism between factor modules over commutative semi-simple ring

Let $R$ be a commutative semi-simple ring with unity (i.e. all modules over $R$ is semi-simple) , let $P \le N \le M$ be a chain of $R$ modules such that $M \cong M/N$ ; then is it true that $M \cong M/P$ ?

I can prove a kind of a dual version , that if $R$ is commutative semi-simple ring and $P \le N \le M$ is a chain of $R$ modules such that $P \cong M$ , then $M \cong N$ .

You can prove this using your dual version, for instance. By semisimplicity of the ring, we may fix a complementary submodule $N'$ for $N$ within $M$, as well as a complement $P'$ for $P$ within $N$, so that we have $$N = P \oplus P'$$ and $$M = N \oplus N' = P \oplus P' \oplus N'.$$ But then $N'$ is a submodule of $M$ with $N' \cong M/N \cong M$. It follows from the dual version that $P' \oplus N'$ is also isomorphic to $M$, which means that $$M/P \cong P' \oplus N' \cong M.$$