Are $\mathbb{Z}$ and $\mathbb{Z}_n$ the only rings (with identity) whose modules are equivalent to abelian groups?

Question

Let $R$ be a ring with identity. Let $M$ and $N$ be $R$-modules. Let $f$ be an (arbitrary) group homomorphism from $M$ to $N$. Under what conditions on $R$,$M$, and $N$ is $f$ also a $R$-module homomorphism?

Is $R=\mathbb{Z}$ or $R=\mathbb{Z}_n$ necessary? If so, is there a general method to construct group homomorphisms which are not module homomorphisms?

Thanks!