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Math Help - Isomorphism ring :P

  1. #1
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    Talking Isomorphism ring :P

    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!
    Last edited by GTK X Hunter; December 23rd 2009 at 09:49 AM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    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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