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Math Help - Rings theory....

  1. #1
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    Rings theory....

    Hello everybody, thanks for reading....
    I've got this question that I don't seem to be able to solve on my own :-\
    R is a ring. x is an nilpotent (?) element in it - meaning that there exists some n in N such that x^n = 0 (and x is not 0 of course).

    There were some questions posed, and I got stuck on the last one:
    prove that the element (x+u), where u is comutative with x (meaning xu=ux), is reversible (?) (meaning, that there exists an element r in R such that r*(x+u) = (x+u)*r = 1).
    I try all the tricks up my sleeve and still am stuck.

    a question before I've proved (1+x) is reversible, noticing that:
    (1+x)(1-x+x^2-....+x^n-1) = 1 (since x^n is 0)... but this question seems much harder.

    Thank you very much for reading/responding!
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  2. #2
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    Quote Originally Posted by aurora View Post
    Hello everybody, thanks for reading....
    I've got this question that I don't seem to be able to solve on my own :-\
    R is a ring. x is an nilpotent (?) element in it - meaning that there exists some n in N such that x^n = 0 (and x is not 0 of course).
    If x^n = 0 then x^n + 1 = (x+1)(x^{n-1}-x^{n-2}+...\pm x \mp 1) if n is odd. However, it is safe to assume that n is odd because if x^n = 0 \implies x^{n+1}=0. This shows that 1+x is an invertible element in the ring. The problem asks to show u+x is invertible if u is invertible. Notice that u+x is invertible if and only if u^{-1}(u+x) = 1 + u^{-1}x is invertible. But u^{-1}x is nilpotent because (u^{-1}x)^n = u^{-n}x^n =0. By the above result it means 1+u^{-1}x is invertible and consequently u+x is invertible.
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  3. #3
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    Thank you for the response.
    However, I was not given the fact that u is invertible - only that it commutes with x (ux=xu)....
    :-\

    I'm sorry - I just reread the question and realized I am being given the fact that u is invertible :-)
    Sorry again, thanks!
    Last edited by aurora; November 29th 2008 at 10:25 AM.
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  4. #4
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    Then it's wrong! 0x=x0 , but 0+x=x is nilpotent so not invertible...
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  5. #5
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    Indeed, I've written above - I missed the fact that u is invertible .

    Thank you.
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