# Definition:Division Algebra

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## Definition

Let $\struct {A_F, \oplus}$ be an algebra over field $F$ such that $A_F$ does not consist solely of the zero vector $\mathbf 0_A$ of $A_F$.

### Definition 1

$\struct {A_F, \oplus}$ is a **division algebra** if and only if:

- $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$

### Definition 2

$\struct {A_F, \oplus}$ is a **division algebra** if and only if it has no zero divisors:

- $\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$

## Also defined as

Some sources define **division algebra** as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines a **unitary division algebra**.

That is, a **division algebra** is required to have an **identity element**.

## Also known as

A **division algebra** is also known as a **associative division algebra**.

## Also see

- Division Algebra has No Zero Divisors, in which the two definitions are shown to be equivalent.

- Results about
**division algebras**can be found**here**.