1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Abstract Algebra Help Date
Can anyone help me with these abstract algebra proofs? Oct 30, 2012
Abstract algebra help Nov 30, 2011
Abstract algebra HELP, Nov 26, 2010
Abstract Algebra help! Oct 22, 2010
Help with abstract algebra proof Feb 10, 2010