# 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 $a \ne \mathbf 0_R$ and $b \ne \mathbf 0_R$.

Then $a$ and $b$ are zero divisors of $A_R$ if and only if:

$a \oplus b = \mathbf 0_R$