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, . . . , x_n) a multilinear polynomial over C, ρ a nonzero right ideal of R andm > 1 a fixed integer such that ([d(f(r₁, . . . , r_n)), f(r₁, . . . , r_n)])^m = [d(f(r₁, . . . , r_n)), f(r₁, . . . , r_n)] for all r₁, . . . , r_n ∈ ρ. Then either [f(x₁,..., x_n), x_n+1]x_n+2 is an identity for ρ or d(ρ)ρ = 0. [ABSTRACT FROM AUTHOR]