Division algebras with left algebraic commutators

Mehdi Aghabali, S. Akbari, M.H. Bien

Research output: Contribution to journalArticlepeer-review

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 a0 + a1x + ⋯ + anxn (resp. right polynomial a0 + xa1 + ⋯ + xnan) over K such that a0 + a1a + ⋯ + anan = 0 (resp. a0 + aa1 + ⋯ + anan). 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.
Original languageEnglish
Pages (from-to)807-816
JournalAlgebras and representation theory
Volume21
Issue number4
Early online date3 Oct 2017
DOIs
Publication statusPublished - Aug 2018

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