Let $k\ge 2$ and $m_{1},…,m_{k} \in \mathbb{N}$ with $\gcd(m_{i},m_{j}) = 1$ for all $i\ne j$.
Show that $f(x) = (x,…,x)$ defines a ring homomorphism $f: \mathbb{Z}/m\mathbb{Z} \rightarrow \mathbb{Z}/m_{1}\mathbb{Z} \times … \times \mathbb{Z}/m_{k}\mathbb{Z}$ with $m=m_{1}\cdot \cdot \cdot m_{k}$
I am stuck since I don't really see where and how to begin. Therefore I am very thankful for any hints in the right direction.