Annihilator Conditions with Generalized Derivations on Multilinear Polynomials

Abstract

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f(x1,…,xn) a multilinear polynomial over C, a ∈ R, and I a nonzero right ideal of R. Suppose that a[g(f(r1,…,rn)), f(r1,…,rn)]=0 for all ri∈ I and aI ≠ 0. Then either g(x)=a1x with (a1-γ)I=0 for some a1∈ U and γ ∈ C, or there exists an idempotent element e ∈ soc (RC) such that IC=eRC and one of the following holds: (i) f(x1,…,xn) is central-valued in eRe; (ii) g(x)=bx+xc, where b, c ∈ U with (c-b-α)e=0 for some α ∈ C and f(x1,…,xn) is central-valued in eRe.