A note on local commutators in division rings with involution
Bull. Korean Math. Soc. 2019 Vol. 56, No. 3, 659-666
Published online March 4, 2019
Printed May 31, 2019
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
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