On n-generalized commutators and Lie ideals of rings

摘要

Let R be an associative ring. Given a positive integer n >= 2, for a(1), ..., a(n) is an element of 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) is an element of A for all x(i), y(i) is an element of R and all a is an element of 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.