*-Group identities on units of division rings. Issue 7 (18th June 2021)

• Record Type: Journal Article
• Authors: Bien, M. H.; Ramezan-Nassab, M.; Viet, D. H.
• Is Part Of: Communications in algebra. Volume 49:Issue 7(2021)
• Page Start: 3010
• Page End: 3019
• Publication Date: 2021-06-18
• Subjects: Division ring -- Engel group -- group identity -- involution; 16K40 -- 16R50 -- 16U60 -- 16W10; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.1887205
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let D be a division ring with infinite center F . By a well known result of Amitsur, if U ( D ) satisfies a group identity, then D is commutative. Now assume that D has an involution * of the first kind. In this paper, among other results, we show that if U ( D ) satisfies a *-group identity, then either D is commutative or dim F D = 4 and * is of the symplectic type. As a result, let N be a *-invariant normal subgroup of U ( D ) such that all symmetric elements of N are central (this is the case when, for example, each symmetric element of N is bounded Engel). Then either N is central or dim F D = 4 and * is of the symplectic type.