Definition:Division Algebra

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 see

• Equivalence of Definitions of Division Algebra

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

• Results about division algebras can be found here.