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Idempotent elements

  1. Feb 27, 2005 #1
    here's a real tough one ( at least for me) ....show that the ring Z/mnZ where m ,n are relatively prime has an idempotent element other than 0 and 1.
    i looked at examples and it works....
    do we look for solutions of the equation a^2 -a = kmn , for some k in Z( that is, other than 0 and 1)?
  2. jcsd
  3. Feb 27, 2005 #2

    matt grime

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    m and n are coprime. The only thing you know about coprime integers is that there are numbers a and b such that am+bn=1. What can you conclude now?
  4. Feb 27, 2005 #3
    ok...so am + bn= 1 implies......1-bn = am
    that implies... bn(1-bn) = abmn = 0...
    so bn is an idempotent element ....
    i looked at " am +bn =1" a hundred times before posting this question....but it flashed just now! thanks a ton!
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