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Let  R  be a prime ring with its Utumi ring of quotients  U ,    the extended centroid of  R ,  H  a generalized derivation of  R  and  L  a noncentral Lie ideal of  R . Suppose that there exists    such that    for all   , where    is a fixed integer. Then one of the following holds: (i) There exists    such that    for all   , unless  R  satisfies  s 4  . (ii)   ,  R  satisfies  s 4   and there exists an element    such that    for all   . (iii)    and  R  satisfies  s 4  .

Left annihilators of power values of commutators with generalized derivations