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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- September 8th 2009, 07:37 AMCoda202Rings 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. - September 8th 2009, 07:45 AMaman_cc
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.