On n-generalized commutators and Lie ideals of rings.

Let R be an associative ring. Given a positive integer n ≥ 2 , for a 1 , ... , a n ∈ R we define [ a 1 , ... , a n ] n : = a 1 a 2 ⋯ a n − a n a n − 1 ⋯ a 1 , the n -generalized commutator of a 1 , ... , a n . 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 [ x 1 , ... , x r , a , y 1 , ... , y s ] n ∈ A for all x i , y j ∈ 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. [ABSTRACT FROM AUTHOR]