If you mod out $\mathbb{Z}$ be a nontrivial prime power $p^k$, $k>1$, then why can't $\mathbb{Z}/(p^k)=\mathbb{Z}/(n)\oplus\mathbb{Z}/(m)$ for some such submodules?
If that where the case, then $\mathbb{Z}/(n)\cap\mathbb{Z}/(m)=0$, but $mn$ is in both submodules, so $p^k\mid mn$. If say $p^k\mid m$, then $\mathbb{Z}/(m)=0$, so we must have at least $p\mid m$ and $p\mid n$. Is there a problem here? Perhaps they cannot generate $\mathbb{Z}/(p^k)$ any more?