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Math Help - Rings with unity and zero-divisors

  1. #1
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    Rings with unity and zero-divisors

    Let R be a ring with unity

    Prove that if x in R is a unit in R, then x is not a zero-divisor in R.

    Show an example of a ring, R, and an element x in R, such that x is not zero, x is not a unit and x is not a zero-divisor of R.
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  2. #2
    Super Member
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    Aasuming R is commutative. Else you will have to be more specific, I guess.

    x is unit. So there is y, such that x.y=y.x=1

    Let
    x.a = 0
    y.x.a = y.0 = 0
    (y.x).a = 0
    1.a = a = 0

    Hence x can't be zero-divisor.
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