Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

Abstract

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).