This is what I have to prove:
The map $\varphi : m\mathbb Z \to \mathbb Z _{\frac{n}{m}}$ where $m \mid n$ given by $\varphi(x) =\overline{\left(\frac{x}{m}\right)}$ is a (not necessarily unital) ring homomorphism iff $m=1$ or $m=n$.
The backward direction is clear to me. It is the forward direction which I have been unable to prove.
Here's my attempt:
Suppose that the given $\varphi$ is a ring homomorphism. Then $\varphi (m^2)=\varphi (m)^2$, so, $\overline m =\overline 1$. Hence $\frac{n}{m} \mid m-1$.
Further assume that $m \ne n$. Let $p$ be any prime dividing $\frac{n}{m}$.
Then $p$ divides $n$, also, $p$ divides $m-1$. So $p$ cannot divide $m$.
I have not been able to complete. Hints will be appreciated!