Commutativity of prime rings involving generalized derivations.

In this paper we investigate identities with two generalized derivations in prime rings. Let R be a 2-torsion free prime ring admitting two generalized derivations F and G, not both zero. Among others, we prove that if F(xy) + G(yx) 2 Z(R) for all x; y 2 R, then R is a commutative. Also, if the ring R is equipped with an involution of the second kind and F(xx) + G(x+x) 2 Z(R) for all x 2 R, then R is commutative. The proved theorems give a rise to many corollaries which recover well-known results on (generalized) derivations and left multiplier maps on prime rings (resp. with involution). All along the paper, examples are given to discuss the necessity of our assumptions. [ABSTRACT FROM AUTHOR]