# Division Algebras with Left Algebraic Commutators

- 171 Downloads

## Abstract

Let *D* be a division algebra with center *F* and *K* a (not necessarily central) subfield of *D*. An element *a* ∈ *D* is called left algebraic (resp. right algebraic) over *K*, if there exists a non-zero left polynomial *a* _{0} + *a* _{1} *x* + ⋯ + *a* _{ n } *x* ^{ n } (resp. right polynomial *a* _{0} + *x* *a* _{1} + ⋯ + *x* ^{ n } *a* _{ n }) over *K* such that *a* _{0} + *a* _{1} *a* + ⋯ + *a* _{ n } *a* ^{ n } = 0 (resp. *a* _{0} + *a* *a* _{1} + ⋯ + *a* ^{ n } *a* _{ n }). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.

## Keywords

Division algebra Commutators Laurent polynomial identity Maximal subfield Left algebraic## Mathematics Subject Classification (2010)

05A05 05E15 16K20 17A35## Notes

### Acknowledgements

The research of the first author was supported by ERC grant number 320974. The second author is indebted to the Research Council of Sharif University of Technology for support. The third author acknowledges support from Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2016-18-01.

## References

- 1.Albert, A.A., Muckenhoupt, B.: On mtrices of trace zeros. Mich. Math. J.
**4**(1), 1–3 (1957)CrossRefzbMATHGoogle Scholar - 2.Amitsur, S.A.: Rational identities and applications to algebra and geometry. J. Algebra
**3**, 304–359 (1966)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Beidar, K.I., Martindale, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, Inc., New York (1996)zbMATHGoogle Scholar
- 4.Bell, J.P., Drensky, V., Sharifi, Y.: Shirshov’s theorem and division rings that are left algebraic over a subfield. J. Pure Appl. Algebra
**217**, 1605–1610 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Chebotar, M.A., Fong, Y., Lee, P.H.: On division rings with algebraic commutators of bounded degree. Munscripta Math.
**113**, 153–164 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Herstein, I.N. , Scott, W.R.: Subnormal subgroups of division rings. Canad. J. Math.
**15**, 80–83 (1963)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Jacobson, N.: Structure theory for algebraic algebras of bounded degree. Ann. of Math. (2)
**46**(4), 695–707 (1945)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Lam, T.Y.: A First Course in Noncommutative Rings, 2nd edn, GTM, No. 131. Springer, New York (2001)CrossRefGoogle Scholar
- 9.Leung, K.H.: On a generalization of Albert’s Theorem. Israel J. Mth.
**69**(3), 337–350 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Mahdavi-Hezavehi, M., Akbari-Feyzaabaadi, S., Mehraabaadi, M., Hajie-Abolhassan, H.: Commutators in division rings II. Comm. Algebra
**23**(8), 2881–2887 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Rowen, L.H.: Ring Theory, Student Edition. Academic Press, Boston (1991)Google Scholar
- 12.Thompson, R.C.: Commutators in the special and general linear groups. Trans. Am. Math. Soc.
**101**(1), 16–33 (1961)MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.