Define $\mathrm{E}(\mathbb{Z}_{i})$ to be the group of invertible elements of the ring with unity $\mathbb{Z}_{i}$.
Show that $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to $\mathrm{E}(\mathbb{Z}_{m}) \times \mathrm{E}(\mathbb{Z}_{n})$ if and only if $m$ and $n$ are relatively prime.
I see that it is relatively easy to prove that $\mathbb{Z}_{mn}$ is isomorphic to $\mathbb{Z}_{m}\times \mathbb{Z}_{n}$ iff $m$ and $n$ are relatively prime using the Chinese Remainder Theorem. However, is it possible to make an extension of this theorem to prove the above statement?