Isomorphism ring :P

**GTK X Hunter** · December 23rd 2009, 09:38 AM

Suppose $r$ and $s$ are positive integer numbers with $gcd(r,s)=1$ , then show that the mapping $\varphi : Z_{rs}\rightarrow Z_r \times Z_s$ with $\varphi (n) = n (1,1)$ is isomorphism ring!

**Bruno J.** · December 23rd 2009, 10:39 AM

What you mean is that it is a ring isomorphism.

I'll let you show that it's a ring homomorphism. To see that it's an isomorphism, note that $\ker \varphi = \{n \in \mathbb{Z}_{rs} : n(1,1) = (1,1)\} = \{n \in \mathbb{Z}_{rs} : n \equiv 1 \mod r \mbox{ and } n \equiv 1 \mod s\}$ $= \{n \in \mathbb{Z}_{rs} : n \equiv 1 \mod r\} \cap \{ n \equiv 1 \mod s\} = \{1\}$ . (Chinese remainder theorem!) So the map is injective. Since we have $|\mathbb{Z}_{rs}| = |\mathbb{Z}_{r} \times \mathbb{Z}_{s}| = rs$ we are done.

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