Definition:Division Algebra/Definition 2

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 $A_F$.

$A$ is a division algebra 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

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