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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.

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If $M$ is an $R$-submodule of $R_1$ and $N$ is a submodule of $R_2$, then the idempotent $e_1$ acts as identity on $M$ but annihilates all of $N$.

If $\phi:M\to N$ is a morphism of $R$-modules, we have, for all $m\in M$ $$ \phi(m)=\phi(me_1)=\phi(m)e_1=0. $$ So we see that any homomorphism is constant zero.

We could, of course, equally well have used the other idempotent $e_2$ to show that there are no non-trivial homomorphisms of $R$-modules from a submodule of $R_2$ to $R_1$.

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