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

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 ≠2,3 such that the Lie algebra [R] is 5-Engel but the group U(R) is not.