On n-generalized commutators and Lie ideals of rings

TL;DR

This paper explores n-generalized commutators in rings, showing that in noncommutative prime rings, nonzero Lie ideals contain nonzero ideals, extending Herstein's classical results and discussing related generalizations.

Contribution

It extends classical results by proving that nonzero n-generalized Lie ideals contain nonzero ideals in noncommutative prime rings, and discusses related generalizations.

Findings

Nonzero n-generalized Lie ideals contain nonzero ideals in prime rings.

In simple rings, R equals its n-generalized commutator.

The paper extends Herstein's classical results to n-generalized commutators.

Abstract

Let R be an associative ring.In the paper we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n > 2, 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 (Portugal. Math., 1954). Some generalizations and related questions on n-generalized commutators and their relationship with noncommutative polynomials are also discussed.