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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

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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.

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