Definition:Alternating Bilinear Mapping

Definition

Let $R$ be a commutative ring.

Let $\struct {A_R, \oplus}$ be an algebra over $R$.

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

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

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

Characteristic Not $2$

Let $R$ have a characteristic not equal to $2$.

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

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

Also see

• Definition:Alternating Bilinear Form

• Equivalence of Definitions for Alternating Bilinear Mapping on Ring of Characteristic Not 2

• Results about alternating bilinear mappings can be found here.

Sources

• 1965: Serge Lang: Algebra
• John C. Baez: The Octonions (2002): 1.1 Preliminaries