Power values of cocentralizing b-generalized derivations of prime rings

Let Formula: see text be a prime ring, Formula: see text the right maximal ring of quotients of Formula: see text, Formula: see text the extended centroid of Formula: see text, Formula: see text a non-central Lie ideal of Formula: see text, Formula: see text and Formula: see text are non-zero Formula: see text-generalized derivations of Formula: see text. Suppose that there exists a fixed positive integer Formula: see text such that Formula: see text for all Formula: see text. Then one of the following holds: (i) Formula: see text satisfies Formula: see text and Formula: see text; (ii) There exists some Formula: see text such that Formula: see text for all Formula: see text; (iii) There exists some Formula: see text such that Formula: see text for all Formula: see text; (iv) Formula: see text satisfies Formula: see text and there exist Formula: see text such that Formula: see text for all Formula: see text with Formula: see text.