Find all ring-homomorphisms $\displaystyle \Phi: Z_n \rightarrow Z_m$

Thanks!

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- Sep 24th 2008, 05:18 AMroporteRing-Homomorphisms
Find all ring-homomorphisms $\displaystyle \Phi: Z_n \rightarrow Z_m$

Thanks! - Sep 24th 2008, 05:31 AMThePerfectHacker
First, $\displaystyle [1]_n\mapsto [1]_m$ by definition - (by definition for ring homomorphism for commutative rings).

A ring homomorphism is also a group homomorphism between $\displaystyle (\mathbb{Z}_n,+)$ and $\displaystyle (\mathbb{Z}_m,+)$. Since $\displaystyle [1]_n$ generates the group it means this homomorphism is completely determined by its value on this element. Can you finish it now?