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