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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?

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    $\begingroup$ In $\mathbb{Z}$, every element but $\pm 1$ is neither a unit nor a zero divisor. $\endgroup$
    – Alex Wertheim
    Dec 14, 2014 at 15:38

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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.

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