An Engel condition with two generalized derivations on Lie ideals of prime rings.

Let R be a prime ring, let L be a noncentral Lie ideal of R and let g , h be two generalized derivations of R. In this paper, we characterize the structure of R and all possible forms of g and h such that [ g (x m) x n − x s h (x t) , x r ] k = 0 for all x ∈ L , where m , n , s , t , r , k are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167]. [ABSTRACT FROM AUTHOR]