Siberian Federal University

n the present article, we characterize generalized derivations and left multipliers of prime rings involving commutators with idempotent values. Precisely, we prove that if a prime ring of charac- teristic different from 2 admits a generalized derivation G with an associative nonzero derivation g of R such that [G(u); u]n = [G(u); u] for all u 2 f[x; y] : x; y 2 Lg; where L a noncentral Lie ideal of R and n > 1 is a fixed integer, then one of the following holds: (i) R satisfies s4 and there exists 2 C; the extended centroid of R such that G(x) = ax + xa + x for all x 2 R; where a 2 U; the Utumi quotient ring of R; (ii) there exists 2 C such that G(x) = x for all x 2 R: As an application, we describe the structure of left multipliers of prime rings satisfying the condition ([Tm(u); u])n = [Tm(u); u] for all u 2 f[x; y] : x; y 2 Lg; where m; n > 1 are fixed integers. In the end, we give an example showing that the hypothesis of our main theorem is not redundant