Identities with generalized derivations on multilinear polynomials in prime rings

摘要

Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R,C = Z(U) the extended centroid of R, and f (x_1, . . . , x_n) be amultilinear polynomial over C, not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that F( f (x_1, . . . , x_n))G( f (x_1, . . . , x_n))+G( f (x_1, . . . , x_n))F( f (x_1, . . . , x_n)) = 0 for all x_1, . . . , x_n ∈ R. Then f (x_1, . . . , x_n)_2 is central valued on R and one of the following holds.