3
$\begingroup$

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?

  • 2
    Yes: just show that if $R$ and $S$ are rings, then the units of $R\times S$ are exactly the products of units of $R$ and units of $S$. Also, if you address the comment (start it with `@whoever`) then the recipient gets a notification. If it's your question or your answer you get notified anyway, but not otherwise. Finally, if you figure it out, consider writing it up as an answer; that way others can comment and make suggestions, and eventually you can accept your own answer.2011-01-03

0 Answers 0