Malcev ring

This is a variation of Lie ring|Find other variations of Lie ring |

Definition

A Malcev ring (also called a Moufang-Lie ring) is a non-associative ring (i.e., a not necessarily associative ring) $M$ with multiplication $*$ that satisfies the following two conditions:

• The ring is an alternating ring: $\!x*x=0\ \forall \ x\in M$
• The multiplication satisfies the Malcev identity, i.e., the following is true for all $x,y,z\in M$:

$\!(x*y)*(x*z)=(((x*y)*z)*x)+(((y*z)*x)*x)+(((z*x)*x)*y)$

Relation with other properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lie ring