Power Values of Derivations on Multilinear Polynomials in Prime Rings

Abstract

Let R be a prime ring with center Z(R) and with extended centroid C, d a derivation of R and f(x1,…,xn) a nonzero multilinear polynomial over C, m ≥ 1 and p ≥ 1 two integers. In the present paper, we study the situations (i) ((d(f(x1,…,xn)))m − f(x1,…,xn))p = 0; (ii) ((d(f(x1,…,xn)))m − f(x1,…,xn))p ∈ Z(R) for all x1,…,xn in some subsets of R.