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Nilpotent elements in a ring

  1. Oct 6, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that 0 is the only in R if and only if a^2 = 0 implies a = 0.

    2. Relevant equations
    none

    3. The attempt at a solution
    So I'm not sure if I'm doing this right.
    a^2 = a*a = 0. Therefore, either a or a is zero.

    The reason i'm not sure about this is because i'm thinking about matrices, where matrix A^2 can equal zero while A doesn't equal zero.

    Also, did the logic that I use only work if the original question considered the ring a domain?
     
  2. jcsd
  3. Oct 6, 2014 #2

    RUber

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    Homework Helper

    For the if and only if, you should have to demonstrate the proof both ways. If there is a unique zero, then ##a^2=0 \implies a=0##, and if ##a^2=0 \implies a=0##, then zero is unique.
     
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