An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Abstract

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x 1,…, x n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r 1,…, r n ∈ R, a[F 2(f(r 1,…, r n )), f(r 1,…, r n )] = 0, then one of the following statements hold: 1. a = 0; 2. There exists λ ∈C such that F(x) = λx, for all x ∈ R; 3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c 2 ∈ C; 4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c 2 ∈ C.