Left Annihilator of Identities Involving Generalized Derivations in Prime Rings

ABSTRACT. Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R and 07 a ER. If R admits a generalized derivation F such that a(F(uz) + F(u)2) = 0 for all u E L, then one of the following holds: (1) there exists b EU such that F(x) = bx for all x ER, with ab = 0; (2) F(x) = 7x for all x ER; (3) char (R) = 2 and R satisfies 84; (4) char (R) + 2, R satisfies 84 and there exists b EU such that F(x) =bx for all x ER We also study the situations (i) a(F(xmyn) + F(um)F(yn)) = 0 for all x, y ER, and (ii) a(F(xmy") + F(y")F(x)) = 0 for all x,y e R in prime and semiprime rings.