Alternating ring

Definition
An alternating ring is a set $$R$$ equipped with binary operations $$+,\cdot$$, a unary operation $$-$$, and a constant $$0$$, such that:


 * 1) $$(R,+,0,-)$$ is an abelian group.
 * 2) The following two distributivity laws hold for all $$x,y,z \in R$$:
 * 3) * $$x \cdot (y + z) = x \cdot y + x \cdot z$$
 * 4) * $$(x + y) \cdot z = x \cdot z + y \cdot z$$
 * 5) For any $$x \in R$$, $$x \cdot x = 0$$.

Note that, from these axioms, we can deduce that $$x \cdot y + y \cdot x = 0$$ for all $$x,y \in R$$.

Stronger structures

 * Weaker than::Lie ring

Weaker structures

 * Stronger than::Ring