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 $*$:

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

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