Let $m$ and $n$ be positive integers. I want to know when $\mathbb Z_n$ is a semisimple $\mathbb Z_m$-module. I do know that $\mathbb Z_n$ is a $\mathbb Z_m$-module if and only if $n$ is a factor of $m$.

Any leading answer would be appreciated of course!

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    $\begingroup$ What do you mean by $\mathbb{Z}_m$? $m$-adic numbers or $\mathbb{Z}/m\mathbb{Z}$? $\endgroup$ – Ying Zhou Mar 5 '18 at 0:00

When $n$ is a factor of $m$, the structure of $\mathbb{Z}_n$ as a module over $\mathbb{Z}_m$ is the same as a module over $\mathbb{Z}$ as far as submodules are concerned.

What are the semisimple $\mathbb{Z}$-modules? When is a cyclic group semisimple?


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