Equivalent Definition for Alternating Bilinear Mapping

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Theorem

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$ with the property that $\operatorname{Char}\left({R}\right) \neq 2$.

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 for all $a \in A_R$, $a \oplus a = 0$. Then for all $u, v \in A_R$:

\(\displaystyle 0\) \(=\) \(\displaystyle \left({u + v}\right) \oplus \left({u + v}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle u \oplus \left({u + v}\right) + v \oplus \left({u + v}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle u \oplus v + v \oplus u = 0\)


$\Box$

Let $\oplus$ be a bilinear map with the property that for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$.

Let $u \in A_R$.

Then:

\(\displaystyle 0\) \(=\) \(\displaystyle u \oplus u + u \oplus u\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \left({u \oplus u}\right)\)

Since the characteristic of $R$ is not $2$, thus:

$u \oplus u = 0$


$\blacksquare$