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

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

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

That is, such that $x$ is either a left zero divisor or a right zero divisor.

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 run the words together: zerodivisor.

Some use the more explicit and pedantic divisor of zero.

Also defined as
Some sources define a zero-divisor as an element $x \in R^*$ such that:
 * $\exists y \in R^*: x \circ y = 0_R$

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

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

That is, the element $0_R$ itself is not classified as a zero divisor.

This definition is the same as the one given on this website as Proper Zero Divisor.

Also see

 * Definition:Left Zero Divisor
 * Definition:Right Zero Divisor


 * Definition:Proper Zero Divisor