Prove that $$\mathbb Z_m/\langle \overline{n}\rangle \cong\mathbb Z_{\text{gcd}(m,n)}.$$ for any $m,n\in \mathbb{N}$.

For this, I know I must show that in $\mathbb Z_m$, $\langle \overline{n} \rangle$ and $\langle \text{gcd}(m,n)\rangle$ are the same ideal, but I'm not sure how I could possibly show that. Any suggestions?


Hint: Consider the projection $\Bbb Z_m \to \Bbb Z_{gcd(m,n)}$ and the first isomorphism theorem.

| cite | improve this answer | |
  • $\begingroup$ So I want to show that it maps one-to-one and onto, correct? $\endgroup$ – Brian Nov 11 '13 at 2:13
  • $\begingroup$ Yes for the 'onto', no for the 'one-to-one', note that if $d = gcd(m,n)$, then $d$ projects to $0$. $\endgroup$ – Vinicius M. Nov 11 '13 at 2:17
  • $\begingroup$ How do I show that it is onto? I know that it maps onto S provided its image is all of S, but I don't know how to prove this for a problem like this. $\endgroup$ – Brian Nov 11 '13 at 3:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.