Let $R$ be a ring.
I found there are two distinct definitions:
Wikipedia Definition
$a\in R$ is a zero divisor iff there exists nonzero $b\in R$ such that $ab=0$ or $ba=0$.
Another:
Proof wiki
nonzero $a\in R$ is a zero divisor iff there exists nonzero $b\in R$ such that $ab=0$ or $ba=0$.
Which one is the usual definition?
The only difference here is whether you consider the zero element to be a zero divisor, which varies greatly based on the author's preference. For example, Atiyah & Macdonald considers zero to be a zero-divisor but Dummit & Foote does not.