Definition:Zero Divisor/Ring

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

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


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

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

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:
 * $x \mathop \backslash 0_R$

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.

Also known as
Some sources hyphenate, as: zero-divisor.

Some use the more explicit and pedantic divisor of zero.