Noncommutative higher commutators

Abstract

Let A be a unital algebra over a field of characteristic different from 2. The main goal of this work is to characterize higher commutators of A. In particular, we show that if H is a noncommutative higher commutator of A and contains the unity, then H is equal to either A or [A,A]. As a result, we prove that if there exist x,y∈A such that 1=[x,y], then the only higher commutators of A are A and [A,A]. We also consider the structure of noncommutative Lie ideals of the form [U,A], where U is a Lie ideal of A, and show that if L=[U,A] is noncommutative and contains the unity, then L=[A,A] and [A,A]⊆U.