Skew-commutative ring

Statement
Let $$R$$ be a non-associative ring (i.e., a not necessarily associative ring). We say that $$R$$ is skew-commutative or anticommutative if, for all $$x,y \in R$$, we have:

$$\! (x * y) + (y * x) = 0$$

where $$*$$ is the multiplicative operation of $$R$$.

Note that when $$2$$ is invertible in $$R$$, skew-commutativity is equivalent to being an alternating ring. When $$R$$ has characteristic two, skew-commutativity is equivalent to being a commutative ring.