A Note on Generalized Derivations of Order 2 and Multilinear Polynomials in Prime Rings

Let R be a prime ring of char(R)≠ 2, U its Utumi ring of quotients and center C = Z(U) its extended centroid, I a both sided ideal of R, f(x₁,…,x_n) a multilinear polynomial over C, that is noncentral-valued on R, F, G be two generalized derivations of R and d be a derivation of R. Let f(I) be the set of all evaluations of the multilinear polynomial f(x₁,…,x_n) in I. If @@@ for all u ∈ f(I), then all possible forms of the maps are determined. As an application of this result, we also study the commutator identity [F²(u)u,G²(v)v] = 0 for all u,v ∈ f(I), where F and G are two generalized derivations of R.