Lie ring

A ring $A$ that satisfies the conditions

$$a^2=0$$

and

$$(ab)c+(bc)a+(ca)b=0$$

(the Jacobi identity), where $a,b,c$ are any elements of $A$. The first of these conditions implies that $A$ is anti-commutative:

$$ba=-ab.$$

The Lie rings form a variety of rings, in general non-associative. It contains, however, all rings with zero multiplication.

See also Non-associative rings and algebras.