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I've been given the following problem as homework:

If $\mathbb(m, n)\neq 1$, prove that $\mathbb{Z}_{mn}$ is not isomorphic to $\mathbb{Z}_{m}\times\mathbb{Z}_{n}$ (here $(m, n)$ = greatest common divisor of $m$ and $n$.)

How do I approach this problem? I think I can use the fact that there is some $c$ such that $$am + bn = c,\ c > 1,$$ but am not sure how to prove this.

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HINT: Look at these simply as additive groups. $\Bbb Z_{mn}$ has an element of order $mn$. Show if $k=\operatorname{lcm}(m,n)$, then $kg=\langle 0,0\rangle$ for each $g\in\Bbb Z_m\times\Bbb Z_n$.

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Hints: You look for a property of one ring that the other ring does not have, and you make sure that the property you look at is invariant under isomorphisms. For instance, you can consider $\mathbb Z_{mn}$ and $\mathbb Z_m\times \mathbb Z_n$ as abelian groups and show these are non-isomorphic (that would suffice since any ring isomorphism will induce a group isomorphism on the underlying abelian groups). You can do that by showing that $\mathbb Z_m\times \mathbb Z_n$ is not cyclic (since you can easily compute the orders of elements in $\mathbb Z_m\times \mathbb Z_n$, by a theorem in group theory you probably saw).

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  • $\begingroup$ Not actually familiar with group theory, or abelian groups. I'll do some research though! $\endgroup$ – Casey Patton Feb 3 '13 at 5:38
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    $\begingroup$ It is probably not a good idea to start with ring theory before you saw at least a little bit of abelian groups. $\endgroup$ – Ittay Weiss Feb 3 '13 at 5:39

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