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(x 1, . . . , x n ) a multilinear polynomial over C, ρ a nonzero right ideal of R and m > 1 a fixed integer such that $\qquad \left ([d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})]\right )^{m}=[d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})] $ for all r 1, . . . , r n ∈ρ. Then either [f(x 1,…,x n ),x n+1]x n+2 is an identity for ρ or d(ρ)ρ = 0.