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Math Help - Ring Proof Problem

  1. #1
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    Ring Proof Problem

    Here's the question:

    An element \displaystyle{a} in a non-trivial ring R is called nilpotent if a^k = 0 for some integer k \geq 1. Prove that if a is nilpotent then a-1 has a multiplicative inverse. Give an example to illustrate this in Z_{18}.

    I've absolutely no idea how to start this question. I'd be very grateful for any help
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  2. #2
    Senior Member roninpro's Avatar
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    You can handle this problem using something of a dirty trick. From the real numbers, we have the following formula:

    \frac{1}{1-a}=1+a+a^2+\ldots+a^{k-1}+a^k+\ldots

    This suggests that we try to show that (1-a)^1=1+a+a^2+\ldots+a^k. Try carrying it out to see what you get.

    Good luck.
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  3. #3
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    Should that be (1-a)^{-1}?
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