On Weakly Locally Finite Division Rings
- 39 Downloads
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.
KeywordsDivision rings Weakly locally finite Gelfand-Kirrilov dimension Linear groups
Mathematics Subject Classification (2010)16K40 16P90
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.
- 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 Research Note, University of California, Berkeley unpublished mimeographed notes (1978)Google Scholar
- 5.Draxl, P.K.: Skew Fields. London Math. Soc. Lecture note series, vol. 81. Cambridge University Press, Cambridge (1983)Google Scholar
- 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.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
- 14.Lam, T.Y.: A First Course in Non-Commutative Rings GTM, 2nd ed., vol. 131. Springer-Verlag, New York (2001)Google Scholar
- 17.McConnell, J.C., Wadsworth, A.R.: Locally PI but not PI division rings of arbitrary GK-dimension. arXiv:1803.09423v1 [math. RA]
- 20.Scott, W.R.: Group Theory. Dover Publication, Inc., New York (1987)Google Scholar
- 22.Smoktunowicz, A.: Some results in noncommutative ring theory. Inter. Congress Math. II, 259–269 (2006), Eur. Math. Soc Zurich (2006)Google Scholar