Definition:Unitary Division Algebra

Definition
Let $\struct {A_F, \oplus}$ be a division algebra.

Then $\struct {A_F, \oplus}$ is a unitary division algebra it has an identity element $1_{A_F}$ called a unit for $\oplus$, that is:
 * $\exists 1_{A_F} \in A_F: \forall a \in A_F: a \oplus 1_{A_F} = 1_{A_F} \oplus a = a$

The unit is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the division algebra.

Variants
Some sources use this as the definition of division algebra.

That is, its unitary nature is subsumed.