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Unit Proof

  1. Sep 30, 2008 #1
    1. The problem statement, all variables and given/known data

    Let R be a ring and x in R such that x^n=0 for some n show that 1 + x is a unit.

    I know then that x is a zero divisor and I need to find y such that y(1+x) = 1.
    I can see in examples that this works and I can prove it for Z mod n. I cant figure out how to prove it for any ring. Please help
    Thanks
     
  2. jcsd
  3. Sep 30, 2008 #2

    HallsofIvy

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    If there exist n such that xn, there exist a smallest such n. Assume, without loss of generality that n is the smallest number such that xn= 0. If n= 1, then x= 0, x+1= 1 which is a unit. If n> 1, then xn-1 is not 0. Let u= xn-1. Then u(1+ x)= xn-1+ xn= xn-1= u. Does that lead anywhere? In particular is u a unit?
     
  4. Sep 30, 2008 #3

    Dick

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    You probably know that you can write 1/(1+x) formally as a power series, 1-x+x^2-x^3+... If x^n=0, that series terminates. Can you show that it's true that (1+x)*(1-x+x^2-...x^(n-1))=1?
     
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