Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(3): 659-666

Online first article March 4, 2019      Printed May 31, 2019

https://doi.org/10.4134/BKMS.b180476

Copyright © The Korean Mathematical Society.

A note on local commutators in division rings with involution

Mai Hoang Bien

University of Science-VNUHCM

Abstract

In this paper, we consider a conjecture of I. N. Herstein for local commutators of symmetric elements and unitary elements of division rings. For example, we show that if $D$ is a finite dimensional division ring with involution $\star$ and if $a\in D^*=D\backslash\{0\}$ such that local commutators $axa^{-1}x^{-1}$ at $a$ are radical over the center $F$ of $D$ for every $x\in D^*$ with $x^\star=x$, then either $a\in F$ or $\dim_{F}D= 4$.

Keywords: division ring, involution, symmetric element, unitary element, local commutator

MSC numbers: 16K20, 16R50

Supported by: This research was funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number C2018-18-03