$r \mathbb{Z}_n$ is projective as $\mathbb{Z}_n$-module

Question

Let $n=rs$ be a natural bigger than $1$, show that $r \mathbb{Z}_n$ is a projective $\mathbb{Z}_n$-module.

Well, when $r$ and $s$ are coprimes we have that $\mathbb{Z}_n \simeq r \mathbb{Z}_n \oplus s\mathbb{Z}_n $. Since $\mathbb{Z}_n$ is projective over itself we have that $r\mathbb{Z}_n$ projective.

How it works when $r$ and $s$ are not coprimes??

Answer 1

It's not true in general. If $r=s=2$, $n=4$, then $2\mathbb{Z}_4$ is not projective as a $\mathbb{Z}_4$-module. To see this, note $\mathbb{Z}_4$ is local and hence projective modules are free, but $2\mathbb{Z}_4$ is not free.