# Definition:Zero Divisor

## Definition

### Rings

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:

- $x \divides 0_R$

### Commutative Rings

The definition is usually made when the ring in question is commutative:

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

The expression:

**$x$ is a zero divisor**

can be written:

- $x \divides 0_R$

### Algebras

Let $\struct {A_R, \oplus}$ be an algebra over a ring $\struct {R, +, \cdot}$.

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

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

Then $a$ is a **zero divisor of $A_R$** if and only if:

- $a \oplus b = \mathbf 0_R$

## Also defined as

Some sources define a **zero divisor** as an element $x \in R_{\ne 0_R}$ such that:

- $\exists y \in R_{\ne 0_R}: x \circ y = 0_R$

where $R_{\ne 0_R}$ 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 a **proper zero divisor**.

## Also known as

Some sources hyphenate, as: **zero-divisor**.

Some sources run the words together: **zerodivisor**.

Some use the more explicit and pedantic **divisor of zero**.

## Warning

Beware the terminology **divisor of zero**.

It is easy to confuse this with the fact that, for every element $a$ of a ring $R$, from Ring Product with Zero we have that $a \circ 0_R = 0_R$.

Hence one may say that *every* such element is a **divisor of zero**.

However, the concept of a zero divisor specifically requires that the $b$ in $a \circ b = 0_R$ is *not* zero.

## Also see

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