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Homework Help: Ring of Integers Isomorphism Problem

  1. Sep 27, 2008 #1
    The problem statement, all variables and given/known data
    Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB.

    The attempt at a solution
    I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to conclude, since ZN and ZA x ZB have the same cardinality and f is injective, that f is surjetive? In any case, given an (a, b) in ZA x ZB, I've been trying to find an n such that f(n) = (a, b) without success. Any tips?
  2. jcsd
  3. Oct 1, 2008 #2


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    Science Advisor
    Homework Helper

    Recall the Chinese remainder theorem.
  4. Oct 1, 2008 #3
    Good tip. Thanks.
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