Suppose $M$ and $M'$ are two $R$-moules (I am most interested in the case of $R$ a DVR). If $M\otimes M'$ is a semi-simple module (i.e., every submodule is a direct summand) then is it true that the tensor factors $M$ and $M'$ are semi-simple ?


I.e., if this is not true then it would be nice to see a counter-example.



If a tensor product of modules is semi-simple, are the tensor factors semi-simple ?
No; if $R$ is a DVR with uniformizing paramenter $t$, the $R$-module $(R/tR) \otimes (R/t^2R) = R/tR$ is semisimple, but $R/t^2R$ is not.


