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Commutative Ring Problem

  1. Jun 4, 2006 #1

    AKG

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    Let S be a ring such that for all s in S, s2 = s. Prove that S is commutative.

    I've proved that for all s and t in S, (s + t)2 = s2 + t2, and also that s + s = 0. How would I go about proving that for all s and t, st = ts? Thanks. By the way, this isn't exactly homework, I was just practicing for the GREs.
     
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  3. Jun 4, 2006 #2

    Hurkyl

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    What (else) is (s + t)²?
     
  4. Jun 4, 2006 #3

    AKG

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    s + t = (s + t)2 [by hypothesis]
    s + t = s2 + st + ts + t2 [expanding]
    s + t = s + st + ts + t [by hypothesis]
    0 = st + ts [cancelling]
    -st = ts [cancelling]
    st = ts [since st + st = 0]

    Thanks.
     
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