Power cocentralizing generalized derivations on left ideals of prime rings

Let R be a prime ring with center Z(R), let L be a nonzero left ideal of R and let g, h be two generalized derivations of R. In this paper, we completely characterize the structure of L and all possible forms of g, h such that g(x(m))x(n) - x(n)h(x(m)) is an element of Z(R) for all x is an element of L, where m, n 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].