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.


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 $R_1$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.