Definition:Zero Divisor/Ring

Definition
Let $\struct {R, +, \circ}$ be a ring.

A zero divisor (in $R$) is an element $x \in R$ such that either:
 * $\exists y \in R^*: x \circ y = 0_R$

or:
 * $\exists y \in R^*: y \circ x = 0_R$

where $R^*$ is defined as $R \setminus \set {0_R}$.

That is, such that $x$ is either a left zero divisor or a right zero divisor.

The expression:
 * $x$ is a zero divisor

can be written:
 * $x \divides 0_R$

Also see

 * Definition:Left Zero Divisor
 * Definition:Right Zero Divisor


 * Definition:Proper Zero Divisor
 * Definition:Regular Element of Ring
 * Definition:Proper Element of Ring