As we know, there's a structure theorem for modules over $\mathbb{Z}$. What happens over $\mathbb{Z}^2$, or more generally, over $\mathbb{Z}^n$? Obviously, these rings are not integral domains, but I wonder if there are some results concerning the structure of modules over them.

Thanks Alex


Whenever you have two unital rings $R$ and $S$, then $e_R=(1,0)$ and $e_S=(0,1)$ are central idempotents in $R\times S$ and every module $M$ decomposes as $e_RM\oplus e_SM$. Now $e_RM$ is an $R$-module and $e_SM$ is an $S$-module.

Then in your case you can apply the structure theorem.

For more detailed proofs, see e.g. Ringel, Schröer, Proposition 12.16.


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.