Alternating loop ring

Definition
An alternating loop ring is a set $$L$$ endowed with binary operations $$+$$ (addition) and $$*$$ (multiplication) and a constant $$0$$ such that:


 * $$L$$ is an abelian loop under $$+$$ with identity element $$*$$.
 * $$*$$ is alternating, i.e., $$a * a = 0$$ for all $$a \in L$$.
 * $$L$$ satisfies both the left and right distributivity relations between $$+$$ and $$*$$:

$$a * (b + c) = (a * b) + (a * c) \ \forall \ a,b,c \in L$$

$$(a + b) * c = (a * c) + (b * c) \ \forall \ a,b,c \in L$$