A 5-Engel associative algebra whose group of units is not 5-Engel

TL;DR

This paper constructs an associative algebra over a field with a 5-Engel Lie algebra of its elements, yet its group of units is not 5-Engel, showing a divergence from previous nilpotency results.

Contribution

It provides a counterexample demonstrating that a 5-Engel Lie algebra does not necessarily imply a 5-Engel group of units in associative rings.

Findings

Constructed an algebra with a 5-Engel Lie algebra

Showed the group of units is not 5-Engel

Counterexample to a generalization of previous results

Abstract

Let R be an associative ring with unity and let [R] and U(R) denote the associated Lie ring (with [a,b]=ab-ba) and the group of units of R, respectively. In 1983 Gupta and Levin proved that if [R] is a nilpotent Lie ring of class c then U(R) is a nilpotent group of class at most c. The aim of the present note is to show that, in general, a similar statement does not hold if [R] is n-Engel. We construct an algebra R over a field of characteristic different from 2 and 3 such that the Lie algebra [R] is 5-Engel but U(R) is not a 5-Engel group.