# $\mathbb Z_n$ as a semisimple $\mathbb Z_m$-module

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 $m$.

• What do you mean by $\mathbb{Z}_m$? $m$-adic numbers or $\mathbb{Z}/m\mathbb{Z}$? – 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?