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

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 for all $a \in A_R$, $a \oplus a = 0$
 * $\oplus$ is an alternating bilinear mapping 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 $u, v \in A_R$:

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:

Because the characteristic of $R$ is not $2$:
 * $u \oplus u = 0$