# Division Algebras with Left Algebraic Commutators

## 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.

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