# Ring-Homomorphisms

• September 24th 2008, 05:18 AM
roporte
Ring-Homomorphisms
Find all ring-homomorphisms $\Phi: Z_n \rightarrow Z_m$

Thanks!
• September 24th 2008, 05:31 AM
ThePerfectHacker
Quote:

Originally Posted by roporte
Find all ring-homomorphisms $\Phi: Z_n \rightarrow Z_m$

First, $[1]_n\mapsto [1]_m$ by definition - (by definition for ring homomorphism for commutative rings).

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