How to show that $\mathbb Z\_{mn}$ is isomorphic to $\mathbb Z\_m \times\mathbb Z\_n$ when $m$ and $n$ are coprime? It is easy to show that the natural map from $\mathbb Z\_{mn}$ to $\mathbb Z\_m \times\mathbb Z\_n$ is a ring homomorphism. How to show that it is bijective?