LIE IDEALS OF SIMPLE RINGS

Let R be a ring. If we replace the original associative product of R whit their canonic Lie product, or [a,b] = ab-ba for every a,b in R, then R would be, a Lie ring. With this new product the additive commutator subgroup of R or [R,R] is a Lie subring of R. Herstein has shown that in a simple ring R with char R≠2, any Lie ideal of R either is contained in Z(R), the center of R or contains [R,R]. Here, besides a new proof of this results, using the matrix structure of a simple ring, we prove that "Any finitely generated Z-module lie ideal of a simple ring with characteristic than 2, is central.

Keywords:

Lie ideal , Division ring , Simple ring