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Abstract Algebra Help! Rings

  1. Feb 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Let R be a ring and a,b be elements of R. Let m and n be positive integers. Under what conditions is it true that (ab)^n = (a^n)(b^n)?



    2. Relevant equations



    3. The attempt at a solution


    We must show ab = ba.

    Suppose n = 2.

    Then (ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = (aa)(bb) = (a^2)(b^2).

    I am not sure where to go from here...
     
  2. jcsd
  3. Feb 20, 2012 #2

    jbunniii

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    First of all, you said "let m and n be integers," but m was never mentioned again. Is there a typo somewhere?

    Second: You said "We must show ab = ba". But then you ASSUMED it was true in this step: a(ba)b = a(ab)b. This is not necessarily true unless the ring is commutative, in which case this problem becomes trivial.

    I would start with the n = 2 case. What must be true in order for (ab)^2 to equal a^2 b^2? This is equivalent to writing

    abab = aabb

    or

    a(ba - ab)b = 0

    Clearly this is true if ab = ba, but is this necessary? What if this is a ring of matrices, for example?
     
    Last edited: Feb 20, 2012
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