Definition:Zero Divisor

Rings
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 et al.

Algebras
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $\left({R, +, \cdot}\right)$.

Let the zero vector of $A_R$ be $\mathbf 0_R$.

Let $a, b \in A_R$ such that $a \ne \mathbf 0_R$ and $b \ne \mathbf 0_R$.

Then $a$ and $b$ are '''zero divisors of $A_R$ iff $a \oplus b = \mathbf 0_R$.

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.