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!
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?