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 different from 2 and 3 such that the Lie algebra [R] is 5-Engel but U(R) is not a 5-Engel group.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2013
- DOI:
- 10.48550/arXiv.1305.1350
- arXiv:
- arXiv:1305.1350
- Bibcode:
- 2013arXiv1305.1350D
- Keywords:
-
- Mathematics - Rings and Algebras;
- Mathematics - Group Theory;
- 16R40;
- 16L30;
- 16N20;
- 17B30;
- 20F45
- E-Print:
- 9 pages, Corollary 1.4 and some references added