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Acta Mathematica Vietnamica

, Volume 44, Issue 2, pp 553–569 | Cite as

On Weakly Locally Finite Division Rings

  • Trinh Thanh DeoEmail author
  • Mai Hoang Bien
  • Bui Xuan Hai
Article

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.

Keywords

Division rings Weakly locally finite Gelfand-Kirrilov dimension Linear groups 

Mathematics Subject Classification (2010)

16K40 16P90 

Notes

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 Information

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

References

  1. 1.
    Akbari, S., Mahdavi-Hezavehi, M.: Normal subgroups of GLn(D) are not finitely generated. Proc. Am. Math. Soc. 128(6), 1627–1632 (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beidar, K.I., Martindale, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, Inc., New York (1996)zbMATHGoogle Scholar
  3. 3.
    Bergman, G.M.: A note on growth functions of algebras and semigroups. Research Note, University of California, Berkeley unpublished mimeographed notes (1978)Google Scholar
  4. 4.
    Besicovitch, A.S.: On the linear independence of fractional powers of integers. J. Lond. Math. Soc. 15, 3–6 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Draxl, P.K.: Skew Fields. London Math. Soc. Lecture note series, vol. 81. Cambridge University Press, Cambridge (1983)Google Scholar
  6. 6.
    Drensky, V.: Free Algebras and PI-Algebras. Graduate Course in Algebra. Springer-Verlag, Singapore (2000)zbMATHGoogle Scholar
  7. 7.
    Deo, T.T., Bien, M.H., Hai, B.X.: On radicality of maximal subgroups in GLn(D). J. Algebra 365, 42–49 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hai, B.X., Deo, T.T., Bien, M.H.: On subgroups in division rings of type 2. Stud. Sci. Math. Hungar. 49(4), 549–557 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hai, B.X., Huynh, L.K.: On subgroups of the multiplicative group of a division ring. Vietnam J. Math. 32(1), 21–24 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Herstein, I.N.: Multiplicative commutators in division rings. Israel J. Math. 31 (2), 180–188 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kharchenko, V.K.: Simple, Prime and Semiprime Rings. In: Handbook of Algebra, vol. 1, pp 761–812. Elsevier North-Holland, Amsterdam (1996)Google Scholar
  12. 12.
    Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. Revised edition. Graduate studies in mathematics. vol. 22, AMS (2000)Google Scholar
  13. 13.
    Kurosh, A.G.: Ringtheoretische probleme, die mit dem Burnsideschen problem über periodische Gruppen in Zusammenhang stehen. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 5, 233–240 (1941)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lam, T.Y.: A First Course in Non-Commutative Rings GTM, 2nd ed., vol. 131. Springer-Verlag, New York (2001)Google Scholar
  15. 15.
    Mahdavi-Hezavehi, M., Mahmudi, M.G., Yasamin, S.: Finitely generated subnormal subgroups of GLn(D) are central. J. Algebra 225(2), 517–521 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mahdavi-Hezavehi, M., Akbari, S.: Some special subgroups of GLn(D). Algebra Colloq. 5(4), 361–370 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    McConnell, J.C., Wadsworth, A.R.: Locally PI but not PI division rings of arbitrary GK-dimension. arXiv:1803.09423v1 [math. RA]
  18. 18.
    Mordell, L.J.: On the linear independence of algebraic numbers. Pac. J. Math. 3, 625–630 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Neumann, B.H.: On ordered division rings. Trans. Am. Math. Soc. 66(1), 202–252 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Scott, W.R.: Group Theory. Dover Publication, Inc., New York (1987)Google Scholar
  21. 21.
    Siegel, C.L.: Algebraische abhäengigkeit von Wurzeln (German). Acta Arith. 21, 59–64 (1972)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Smoktunowicz, A.: Some results in noncommutative ring theory. Inter. Congress Math. II, 259–269 (2006), Eur. Math. Soc Zurich (2006)Google Scholar
  23. 23.
    Zhang, J.J.: On Gel’fand-Kirillov transcendence degree. Trans. Am. Math. Soc. 348(7), 2867–2899 (1996)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zelmanov, E.: Some open problems in the theory of infinite dimensional algebras. J. Korean Math. Soc. 44(5), 1185–1195 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Trinh Thanh Deo
    • 1
    Email author
  • Mai Hoang Bien
    • 1
  • Bui Xuan Hai
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceVNUHCM - University of ScienceHo Chi Minh CityVietnam

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