Power cocentralizing generalized derivations on left ideals of prime rings

Abstract

Let [Formula: see text] be a prime ring with center [Formula: see text], let [Formula: see text] be a nonzero left ideal of [Formula: see text] and let [Formula: see text] be two generalized derivations of [Formula: see text]. In this paper, we completely characterize the structure of [Formula: see text] and all possible forms of [Formula: see text] such that [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers. With this, several known results can be either deduced or generalized. Moreover, our result can be regarded as the one-sided ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317].