Commutators with idempotent values on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f (x1,..., xn) a multilinear polynomial over C, a nonzero right ideal of R and m > 1 a fixed integer such that ([d(f (r1,..., rn), f (r1,..., rn)])m = [d(f (r1,..., rn), f (r1,..., rn)] for all r1,..., rn n. Then either [f (x1,..., xn), xn+1]xn+2 is an identity for or d() = 0. © Indian Academy of Sciences.