# Category:Zero Divisors

This category contains results about **Zero Divisors**.

Definitions specific to this category can be found in Definitions/Zero Divisors.

### 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:

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### E

- Examples of Zero Divisors (4 P)

## Pages in category "Zero Divisors"

The following 8 pages are in this category, out of 8 total.