Generalized derivations preserving Engel condition in prime and semiprime rings

Let [Formula: see text] be a prime ring with [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] its Martindale quotient ring and [Formula: see text] its extended centroid. Let [Formula: see text] and [Formula: see text] be nonzero generalized derivations on [Formula: see text] such that [Formula: see text] Then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text], for any [Formula: see text], unless [Formula: see text], where [Formula: see text] is the algebraic closure of [Formula: see text].

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

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]).

Antiautomorphisms with quasi-generalized Engel condition

Let [Formula: see text] be a ring with 1. Given elements [Formula: see text], [Formula: see text] of [Formula: see text] and the integer [Formula: see text] define [Formula: see text] and [Formula: see text]. We say that a given antiautomorphism [Formula: see text] of [Formula: see text] is commuting if [Formula: see text], all [Formula: see text]. More generally, assume that [Formula: see text] satisfies the condition [Formula: see text] where [Formula: see text], [Formula: see text] are corresponding positive integers depending on [Formula: see text], and [Formula: see text] ranges over [Formula: see text]. To what extent can one say that [Formula: see text] is commuting? In this paper, we answer the question in the affirmative if R is a prime ring containing some idempotent element [Formula: see text]. In the diametrically opposed case in which [Formula: see text] is a division ring the answer is again yes provided [Formula: see text] is algebraic over its center and [Formula: see text] is of finite order. These two major complementary results will be put to work to provide an answer to the general question.

An Engel condition with b-generalized derivations for Lie ideals

Let [Formula: see text] be a prime ring with the extended centroid [Formula: see text], [Formula: see text] a noncommutative Lie ideal of [Formula: see text] and [Formula: see text] a nonzero [Formula: see text]-generalized derivation of [Formula: see text]. For [Formula: see text], let [Formula: see text]. We prove that if [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text] except when [Formula: see text], the [Formula: see text] matrix ring over a field [Formula: see text]. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139–152.]