Definition:Unitary Division Algebra

Theorem
Let $\left({A_F, \oplus}\right)$ be a division algebra.

Then $\left({A_F, \oplus}\right)$ is a unitary division algebra iff it has an identity element $1_A$ called a unit for $\oplus$:
 * $\exists 1_A \in A_R: \forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$

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

Variants
Some sources use this as the definition of division algebra, that is, its unitary nature is subsumed.