Definition:Division Algebra/Definition 1

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 $A_F$.

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

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

That is, for every pair of elements $a, b$ of $A_F$ where $b$ is non-zero, there exists:

a unique element $x$ such that $a = b \oplus x$

a unique element $y$ such that $a = y \oplus b$

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