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Algebras and Representation Theory

, Volume 21, Issue 4, pp 807–816 | Cite as

Division Algebras with Left Algebraic Commutators

  • M. Aaghabali
  • S. Akbari
  • M. H. Bien
Open Access
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Abstract

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element aD is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a 0 + a 1 x + ⋯ + a n x n (resp. right polynomial a 0 + x a 1 + ⋯ + x n a n ) over K such that a 0 + a 1 a + ⋯ + a n a n = 0 (resp. a 0 + a a 1 + ⋯ + a n a n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.

Keywords

Division algebra Commutators Laurent polynomial identity Maximal subfield Left algebraic 

Mathematics Subject Classification (2010)

05A05 05E15 16K20 17A35 

Notes

Acknowledgements

The research of the first author was supported by ERC grant number 320974. The second author is indebted to the Research Council of Sharif University of Technology for support. The third author acknowledges support from Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2016-18-01.

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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of MathematicsThe University of EdinburghEdinburghUK
  2. 2.Department of Mathematical SciencesSharif University of TechnologyTehranIran
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of ScienceHCM-CityVietnam

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