Algebraic commutators with respect to subnormal subgroups in division rings

Abstract

Let \(D\) be a division ring and \(K\) a subfield of \(D\) which is not necessarily contained in the center \(F\) of \(D\). We study the structure of \(D\) under the condition of left algebraicity of certain subsets of \(D\) over \(K\). Among others, it is proved that if \(D^*\) contains a noncentral normal subgroup which is left algebraic over \(K\) of bounded degree \(d\), then \([D:F]\le d^2\). In case \(K=F\), the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of \(D^*\) are algebraic of bounded degree \(d\) over \(f\), then \([D:F]\le d^2\).

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Acknowledgement

The authors would like to express their sincere gratitude to the editor and the referee for their comments and suggestions.

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Correspondence to B. X. Hai.

Additional information

The first and the second author are funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number B2020-18-02.

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Bien, M.H., Hai, B.X. & Trang, V.M. Algebraic commutators with respect to subnormal subgroups in division rings. Acta Math. Hungar. (2021). https://doi.org/10.1007/s10474-020-01109-3

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Key words and phrases

  • algebraic
  • subnormal subgroup
  • division ring
  • maximal subfield

Mathematics Subject Classification

  • 16K20
  • 16K40
  • 16R20
  • 05A05
  • 05E15