An identity on generalized derivations involving multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U, extended centroid C, f(x1,...,xn) a multilinear polynomial over C, which is not central-valued on R and d a nonzero derivation of R. By f(R), we mean the set of all evaluations of the polynomial f(x1,...,xn) in R. In the present paper, we study b[d(u),u]+p[d(u),u]q+[d(u),u]c=0 for all uf(R), which includes left-sided, right-sided as well as two-sided annihilating conditions of the set {[d(u),u]:uf(R)}. We also examine some consequences of this result related to generalized derivations and we prove that if F is a generalized derivation of R and d is a nonzero derivation of R such that <for all uf(R), then there exists aU with a2=0 such that F(x)=xa for all xR or F(x)=ax for all x is an element of R.