Multiplicative groups of Engel associative algebras
by
Galina Deryabina
Department of Computational Mathematics and Mathematical Physics, Bauman Moscow State Technical University, 5 Second Baumanskaya St., 105005, Moscow, Russia
Coauthors: Alexei Krasilnikov

Let R be an associative ring. Let [R] denote the associated Lie ring of R (with [a, b]=ab-ba) and let U(R) denote the multiplicative group of R. It is known that if [R] is nilpotent of class c then the group U(R) is nilpotent of class at most c (Gupta and Levin, 1983) and if [R] is metabelian then U(R) is metabelian (Krasilnikov, 1992, and independently Sharma and Srivastava, 1992). Our main result is as follows.

Theorem. Over each field of characteristic 2 there exists an associative algebra R such that its associated Lie algebra [R] is 3-Engel but its multiplicative group U(R) is not 3-Engel.

Date received: February 22, 2006