Definition:Division Algebra/Definition 2

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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 $A_F$.


$\struct {A_F, \oplus}$ 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

  • Results about division algebras can be found here.


Sources