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.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Alternating ring

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Flexible ring	$x*(y*x)=(x*y)*x$ for all $x,y$	skew-commutative implies flexible