Zero divisor

in a ring or a semi-group with zero element

A non-zero element such that the product with some non-zero element is zero. An element $a$ is called a left (right) divisor of zero if $ab=0$ ($ba=0$) for at least one $b\neq0$.

Comments

Let $A$ be a ring and $M$ a left module over $A$. Then an element $a\neq0$ of $A$ is called a zero divisor in the module $M$ if there is an $m\in M$ such that $am=0$.