On Weakly Locally Finite Division Rings

Abstract

Weakly locally finite division rings were considered in Deo et al. (J. Algebra 365, 42–49, 2012), where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer n ≥ 0 or n = , we construct a weakly locally finite division ring whose Gelfand-Kirillov dimension is n. This fact shows in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Further, for the class of weakly locally finite division rings, we investigate some questions related with the well-known Kurosh Problem and with one of Herstein’s conjectures.

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Notes

  1. 1.

    https://math.stackexchange.com/questions/30687/the-square-roots-of-different-primes-are-linearly-independent-over-the-field-of

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Acknowledgements

The authors would like to thank the referee for his/her comments and suggestions. Also, they sincerely thank Adrian Wadsworth and John McConnell for the conversations by e-mail about McConnell’s construction of examples of locally PI, but not PI division rings of arbitrary GK-dimension.

Funding

This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2016.18.

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Correspondence to Trinh Thanh Deo.

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Deo, T.T., Bien, M.H. & Hai, B.X. On Weakly Locally Finite Division Rings. Acta Math Vietnam 44, 553–569 (2019). https://doi.org/10.1007/s40306-018-0292-x

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Keywords

  • Division rings
  • Weakly locally finite
  • Gelfand-Kirrilov dimension
  • Linear groups

Mathematics Subject Classification (2010)

  • 16K40
  • 16P90