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Some notes on Lie ideals in division rings

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Original languageEnglish
Article number1850049
Number of pages6
JournalJournal of algebra and its applications
Volume17
Issue number03
Early online date7 Apr 2017
DOIs
Publication statusPublished - Mar 2018

Abstract

A Lie ideal of a division ring A is an additive subgroup L of A such that the Lie product [l, a] = laal of any two elements lL, aA is in L or [l, a] ∈ L. The main concern of this paper is to present some properties of Lie ideals of A which may be interpreted as being dual to known properties of normal subgroups of A∗. In particular, we prove that if A is a finite-dimensional division algebra with center F and charF ≠ 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 Mn(A) are also presented.



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