On n -generalized commutators and Lie ideals of rings

Let R be an associative ring. Given a positive integer n ? 2, for a1,...,an R we define [a1,...,an]n:= a1a2?an - anan-1?a1, the n-generalized commutator of a1,...,an. By an n-generalized Lie ideal of R (at the (r + 1)th position with r ? 0) we mean an additive subgroup A of R satisfying [x1,...,xr,a,y1,...,ys]n A for all xi,yj R and all a A, where r + s = n - 1. In the paper, we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n ? 3, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is a noncommutative simple ring, then R = [R,...,R]n. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137-139]. Some generalizations and related questions on n-generalized commutators and their relationship with noncommutative polynomials are also discussed. ? 2022 World Scientific Publishing Company.