Definition:Alternating Bilinear Mapping

Definition

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring.

By definition, $\oplus$ is a bilinear mapping.

Then $\oplus$ is an alternating bilinear mapping if and only if:

$\forall a \in A_R: a \oplus a = 0$

For rings with characteristic other than two, the following condition is equivalent:

$\forall a, b \in A_R: a \oplus b = - b \oplus a$