Definition:Division Algebra

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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 defined as

Some sources define division algebra as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines a unitary division algebra.

That is, a division algebra is required to have an identity element.

Also known as

A division algebra is also known as a associative division algebra.

Also see

  • Results about division algebras can be found here.