# Definition:Division Algebra

## Definition

Let $\left({A_F, \oplus}\right)$ 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

$\left({A_F, \oplus}\right)$ 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

$A$ is a division algebra if and only if it has no zero divisors:

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