[Math] relation between units and non zero divisors in a ring

Question

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?

Answer 1

Units are always non zero-divisors. If $x$ is a unit, then $1=xy$ for some $y\in R$. Now, if $rx=0$, then $r=r(xy)=(rx)y=0$. So $x$ is not a zero-divisor.