Consider a product of rings $R = R_1 \times R_2$. We have that $e_1 = (1,0),e_2 = (0,1)$ are central idempotent elements in $R$.
Prove that $R_1,R_2$ as right $R$-modules don't have non-zero isomorphic submodules.
My thougths (plus Jyrki Lahtonen answer)
I would look at Eric's Wofsey answer to understand why the external product is taking that way.