Thread: 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.

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.