Let $R$ be a ring and $a\in R$.
$a$ is a left zero divisor if there exists $x\in R^*$ such that
\[ax = 0.\]$a$ is a right zero divisor if there exists $x\in R^*$ such that
\[xa = 0.\]$a$ is a zero divisor if it is a left zero divisor or right zero divisor.