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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On free subgroups in division rings
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by Jason P. Bell and Jairo Gonçalves PDF
Proc. Amer. Math. Soc. 148 (2020), 1953-1962 Request permission

Abstract:

Let $K$ be a field, let $\sigma$ be an automorphism, and let $\delta$ be a $\sigma$-derivation of $K$. We show that the multiplicative group of nonzero elements of the division ring $D=K(x;\sigma ,\delta )$ contains a free noncyclic subgroup unless $D$ is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free nonabelian solvable-by-finite groups always contain free noncyclic subgroups.
References
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Additional Information
  • Jason P. Bell
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • MR Author ID: 632303
  • Email: jpbell@uwaterloo.ca
  • Jairo Gonçalves
  • Affiliation: Department of Mathematics, University of Saõ Paulo, Saõ Paulo, SP, 05508-090, Brazil
  • MR Author ID: 75040
  • Email: jz.goncalves@usp.br
  • Received by editor(s): December 13, 2018
  • Received by editor(s) in revised form: September 30, 2019
  • Published electronically: January 29, 2020
  • Communicated by: Jerzy Weyman
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 1953-1962
  • MSC (2010): Primary 12E15, 16K40, 20E05
  • DOI: https://doi.org/10.1090/proc/14888
  • MathSciNet review: 4078080