Noncommutative higher commutators

EROĞLU M. P.

JOURNAL OF ALGEBRA, vol.594, pp.101-108, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 594
  • Publication Date: 2022
  • Doi Number: 10.1016/j.jalgebra.2021.11.034
  • Journal Name: JOURNAL OF ALGEBRA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.101-108
  • Keywords: Lie structure, Commutator, Higher commutator, Lie ideal, Noncommutative algebra, Semiprime algebra
  • Dokuz Eylül University Affiliated: Yes

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 is an element of 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] subset of U.(c) 2021 Published by Elsevier Inc.