It is simple enough to find the $m$ such that $Z_n$ is a projective $Z_m$-module. It is also an easy exercise to show that $Z_n$ is not a projective $\mathbb{Z}$-module. The way I have always looked at this proof is that $Z_n$ is 'too small' to be a projective $\mathbb{Z}$-module (thinking from the use of torsion in the proof).
I am wondering if this is not the best way to internalize this fact as I was told there are 'larger' rings over which $Z_n$ is still projective but I have not found any and rather doubt that this fact is true. Are there 'larger' rings (in the sense that they contain a copy of $\mathbb{Z}$ or $\mathbb{Q}$) over which $Z_n$ is projective?
Note: I mean $Z_n=\mathbb{Z}/n\mathbb{Z}$
If $R$ is any ring whatsoever, $Z_n$ is projective as an $R\times Z_n$-module for the obvious module structure.