On Weakly Locally Finite Division Rings


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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  1. 1.



  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)

    Article  MATH  Google Scholar 

  2. 2.

    Beidar, K.I., Martindale, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, Inc., New York (1996)

    Google 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 Research Note, University of California, Berkeley unpublished mimeographed notes (1978)

  4. 4.

    Besicovitch, A.S.: On the linear independence of fractional powers of integers. J. Lond. Math. Soc. 15, 3–6 (1940)

    MathSciNet  Article  MATH  Google 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)

    Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Herstein, I.N.: Multiplicative commutators in division rings. Israel J. Math. 31 (2), 180–188 (1978)

    MathSciNet  Article  MATH  Google 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)

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Lam, T.Y.: A First Course in Non-Commutative Rings GTM, 2nd ed., vol. 131. Springer-Verlag, New York (2001)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Mahdavi-Hezavehi, M., Akbari, S.: Some special subgroups of GLn(D). Algebra Colloq. 5(4), 361–370 (1998)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Neumann, B.H.: On ordered division rings. Trans. Am. Math. Soc. 66(1), 202–252 (1949)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Smoktunowicz, A.: Some results in noncommutative ring theory. Inter. Congress Math. II, 259–269 (2006), Eur. Math. Soc Zurich (2006)

  23. 23.

    Zhang, J.J.: On Gel’fand-Kirillov transcendence degree. Trans. Am. Math. Soc. 348(7), 2867–2899 (1996)

    Article  MATH  Google Scholar 

  24. 24.

    Zelmanov, E.: Some open problems in the theory of infinite dimensional algebras. J. Korean Math. Soc. 44(5), 1185–1195 (2007)

    MathSciNet  Article  MATH  Google Scholar 

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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.


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

Mathematics Subject Classification (2010)

  • 16K40
  • 16P90