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Homework Help: Algebra Proofs

  1. Feb 25, 2009 #1
    Algebra Proofs!!

    I have two questions just to help verify what I'm doing:

    Let R be a commutative ring with identity. Prove that R is an integral domain if and only if cancellation holds in R (that is, a no equal to 0 and ab=ac in R imply b=c)

    => Suppose cancellation holds: ab=0 -> ab=0a -> a isn't 0 so b=0

    <= Since a doesn't equal 0
    ab=ac -> ab-ac=0 -> a(b-c)=0 -> b-c=0 -> b=c cancellation holds

    alternatively, could the second part be proven as such:

    ab=ac
    a^-1ab=aca^-1
    b=c
    cancellation holds

    My second question is: How do I go about proving the following

    (a^m)(a^n)=a^m+n and (a^m)^n=a^mn
     
  2. jcsd
  3. Feb 26, 2009 #2

    HallsofIvy

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    Re: Algebra Proofs!!

    Your "alternative" cannot be used because a-1 does not necessarily exist!

    For your second question, I would suggest fixing m and using induction on n.
     
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