Equivalent Definition for Alternating Bilinear Mapping

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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 $2$:

$u \oplus u = 0$

$\blacksquare$