Definition:Zero Divisor

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 "$$x|0_R$$", but there is a growing trend to follow the notation above, as espoused by Knuth etc.

Proper Non-Zero Divisor
Note:  Some authors 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 non-zero divisor" is used to define what we call a "zero divisor".