Definition:Proper Zero Divisor

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

A proper zero divisor of $R$ is an element $x \in R^*$ such that:


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

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

That is, it is a zero divisor of $R$ which is specifically not $0_R$.

The presence of a proper zero divisor in a ring means that the product of two elements of the ring may be zero even if neither factor is zero.

That is, if $R$ has proper zero divisors, then $\struct {R^*, \circ}$ is not closed.

Also known as
Some authors exclude $0_R$ as a zero divisor and thus refer to this concept simply as zero divisor.

Also see

 * Definition:Zero Divisor of Ring