# Equivalent Definition for Alternating Bilinear Mapping

Jump to navigation
Jump to search

## Theorem

Let $\struct {A_R, \oplus}$ be an algebra over a ring $R$ with the property that $\Char R \ne 2$.

Then the following definitions for alternating bilinear maps are equivalent:

- $\oplus$ is an alternating bilinear map if and only if for all $a \in A_R$, $a \oplus a = 0$
- $\oplus$ is an alternating bilinear map if and only if for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$

## Proof

Let $\oplus$ be a bilinear map with the property that:

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

Then for all $u, v \in A_R$:

\(\displaystyle 0\) | \(=\) | \(\displaystyle \paren {u + v} \oplus \paren {u + v}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle u \oplus \paren {u + v} + v \oplus \paren {u + v}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle u \oplus v + v \oplus u = 0\) |

$\Box$

Let $\oplus$ be a bilinear map with the property that:

- $\forall a, b \in A_R: a \oplus b + b \oplus a = 0$

Let $u \in A_R$.

Then:

\(\displaystyle 0\) | \(=\) | \(\displaystyle u \oplus u + u \oplus u\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \paren {u \oplus u}\) |

Because the characteristic of $R$ is not $2$:

- $u \oplus u = 0$

$\blacksquare$