Identities involving generalized derivations in prime rings

Abstract

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a nonzero derivation of R and G is a generalized derivation of R. If $$G^2(u)d(u)=0$$ for all $$u\in f(R)$$ , then one of the following holds: