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Nilpotent Elements and Units

  1. Dec 5, 2016 #1
    1. The problem statement, all variables and given/known data
    First Claim: If ##u \in R## is a unit, then it cannot be nilpotent

    Second Claim: If ##u \in R## is nilpotent, then it cannot be a unit

    2. Relevant equations


    3. The attempt at a solution

    I realize these are simple problems, but I have no one to verify my work and I want to be certain I am doing things correctly. Here is a proof of the first claim:

    Suppose the contrary, that ##u## is a unit and also nilpotent. This implies there exists an element ##v## that acts as a multiplicative inverse and a natural number ##n## such that ##u^n = 0##. By the well ordering property, we can take ##n## to be the smallest natural number for which ##u^n=0##. Then

    ##u^n = 0##

    ##u u^{n-1} = 0##

    ##vu u^{n-1} = v0##

    ##u^{n-1} = 0##,

    contradicting the minimality of ##n##.

    Does this seem right? If I am not mistaken, then proof of the second claim is identical.
     
  2. jcsd
  3. Dec 5, 2016 #2

    fresh_42

    Staff: Mentor

    That's correct, although I wouldn't have used ordering and minimality, which looks kind of artificial to me, like a little stone in the shoe.
    You could have simply multiplied by ##v^n## instead.
     
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