# Definition:Zero Divisor/Ring

## Definition

Let $\struct {R, +, \circ}$ 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 $R \setminus \set {0_R}$.

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:

- $x \divides 0_R$

## Notation

The conventional notation for **$x$ is a divisor of $y$** is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by Knuth etc.

From Ronald L. Graham, Donald E. Knuth and Oren Patashnik: *Concrete Mathematics: A Foundation for Computer Science* (2nd ed.):

*The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.*

An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.

Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \ $ or similar to denote non-divisibility.

## 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 $R \setminus \set {0_R}$.

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

- Results about
**zero divisors**can be found here.

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.2$ (in passing) - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 21$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**zero divisor**