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!

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

    User Avatar
    Staff Emeritus
    Science Advisor

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




Similar Discussions: Algebra Proofs
  1. Algebra proof! (Replies: 1)

  2. Algebra proof. (Replies: 4)

  3. Algebra Proof (Replies: 9)

Loading...