Equivalence of Definitions for Alternating Bilinear Mapping on Ring of Characteristic Not 2
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Theorem
Let $R$ be a commutative ring.
Let $\struct {A_R, \oplus}$ be an algebra over $R$.
Let $R$ have a characteristic not equal to $2$.
Then the following definitions for alternating bilinear mappings are equivalent:
- $\oplus$ is an alternating bilinear mapping if and only if for all $a \in A_R$, $a \oplus a = 0$
- $\oplus$ is an alternating bilinear mapping if and only if for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$
Proof
Let $\oplus$ be a bilinear mapping with the property that:
- $\forall a \in A_R: a \oplus a = 0$
Then for all $u, v \in A_R$:
\(\ds 0\) | \(=\) | \(\ds \paren {u + v} \oplus \paren {u + v}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds u \oplus \paren {u + v} + v \oplus \paren {u + v}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds u \oplus v + v \oplus u = 0\) |
$\Box$
Let $\oplus$ be a bilinear mapping with the property that:
- $\forall a, b \in A_R: a \oplus b + b \oplus a = 0$
Let $u \in A_R$.
Then:
\(\ds 0\) | \(=\) | \(\ds u \oplus u + u \oplus u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \paren {u \oplus u}\) |
Because the characteristic of $R$ is not $2$:
- $u \oplus u = 0$
$\blacksquare$