Centralizers and Vanishing Derivations on Multi-linear Polynomials in Prime Rings

Abstract

Let R be a prime algebra over a commutative ring K with characteristic not equal to 2. Let d and δ be non-zero derivations of R, f(x1,…, xn) a multi-linear polynomial over K with n non-commuting variables, and m ≥ 1 a fixed integer. We prove that if δ (d(f(r1,…, rn))m) = 0 for any r1,…, rn ∈ R, then either f(x1,…, xn) is central valued on R or R satisfies the standard identity s4.