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Zero-divisor

Definition

Given a ring $R$, an element $r \in R$ is a zero-divisor if there exists $s \in R$ such that $rs = 0$ or $sr = 0$.

Properties

A zero-divisor is never a unit. (see "unit" page for proof)

If a ring $R$ contains no zero-divisors, we say it is an integral domain.

all these pages adapted with probably insufficient credit from my university's lecture notes