I was writing the proof of the fact that if $R$ and $S$ are semisimple, then so is $R \times S$, where I tried showing that if $M$ is any $R \times S$-module, then M is semisimple. To show this, if we show that $M$ is isomorphic to $M_{1} \oplus M_{2}$, where $M_{1}$ is an R-module and $M_{2}$ is an S-module, then we are done because $R$ is semisimple means $M_{1}$ is semisimple, and likewise $M_{2}$ is semisimple, therefore so is $M$.
However, I couldn't give a proof of the following:
If $M$ is an $R \times S$-module, then $M = M_{1} \oplus M_{2}$ where $M_{1}$ is an $R$-module and $M_{2}$ is an $S$-module.
My attempt: Let $M_1$ be the submodule of $M$ generated by elements of the form $(1,0).m, m \in M$, and $M_2$ be the submodule of $M$ generated by elements of the form $(0,1).m, m \in M$. Then, it is clear that $M = M_{1} + M_{2}$, however I can't see why $M_{1}, M_{2}$ are independent.
I appreciate any help. Thanks!