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 $x,y,z\in R$:
   • $x\cdot (y+z)=x\cdot y+x\cdot z$
   • $(x+y)\cdot z=x\cdot z+y\cdot z$
3. 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 $x,y\in R$.

Relation with other structures

Stronger structures

• Lie ring

Weaker structures

• Ring