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Some notes on Lie ideals in division rings

M. Aaghabali

School of Mathematics, The University of Edinburgh, Edinburgh, UK

E-mail Address: mehdi.aaghabali@ed.ac.uk

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M. Ariannejad

Department of Mathematics, University of Zanjan, Zanjan, Iran

E-mail Address: arian@znu.ac.ir

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, and

A. Madadi

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

E-mail Address: as.madadi@yahoo.com

Corresponding author.

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https://doi.org/10.1142/S0219498818500494Cited by:0 (Source: Crossref)

Abstract

A Lie ideal of a division ring $A$ $A$ is an additive subgroup $L$ $L$ of $A$ $A$ such that the Lie product $[l, a] = l a - a l$ $[l, a] = l a - a l$ of any two elements $l \in L, a \in A$ $l \in L, a \in A$ is in $L$ $L$ or $[l, a] \in L$ $[l, a] \in L$ . The main concern of this paper is to present some properties of Lie ideals of $A$ $A$ which may be interpreted as being dual to known properties of normal subgroups of $A^{*}$ $A^{*}$ . In particular, we prove that if $A$ $A$ is a finite-dimensional division algebra with center $F$ $F$ and $char F \neq 2$ $char F \neq 2$ , then any finitely generated $ℤ$ -module Lie ideal of $A$ is central. We also show that the additive commutator subgroup $[A, A]$ of $A$ is not a finitely generated $ℤ$ -module. Some other results about maximal additive subgroups of $A$ and $M_{n} (A)$ are also presented.

Communicated by L. Rowen

Keywords:

AMSC: 17A01, 17A35

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