Definition:Zero Divisor/Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

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


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

where $R^*$ is defined as $R - \left\{{0_R}\right\}$.

The presence of a 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 zero divisors, then $\left({R^*, \circ}\right)$ is not closed.

The expression:
 * $x$ is a zero divisor

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

The conventional notation for this is $x \mid 0_R$, but there is a growing trend to follow the notation above, as espoused by Knuth et al.

Proper Zero Divisor
Some sources do not insist on $x$ itself being non-zero, that is, zero itself is included in the set of zero divisors.

In this case, the term proper zero divisor is used to define what we call a zero divisor.