Vanishing Derivations and Centralizers of Generalized Derivations on Multilinear Polynomials

Abstract

Let R be a prime algebra over a commutative ring K with unity, and let f(x 1,…, x n ) be a multilinear polynomial over K, not central valued on R. Suppose that d is a nonzero derivation of R and G is a nonzero generalized derivation of R such that for all r 1,…, r n ∈ R. If the characteristic of R is different from 2, then one of the following holds: 1. There exists λ ∈C, the extended centroid of R, such that G(x) = λx, for all x ∈ R; 2. There exist a ∈ U, the Utumi quotient ring of R, and λ ∈C = Z(U) such that G(x) = ax + xa + λx, for all x ∈ R, and f(x 1,…, x n )2 is central valued on R