b-generalized skew derivations acting as 2-Jordan multiplier on multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic not equal to 2, U be its Utumi quotient ring and C be the extended centroid of R. Let ϕbe a multilinear polynomial over C, which is not central valued on R and F, G be two b-generalized skew derivations on R. The purpose of this article is to describe all possible forms of the b-generalized skew derivations F and G satisfying the identity $F(u)u+uG(u)=G(u^2)$, for all u \in {ϕ(ζ) | ζ = (ζ1 . . . , ζn) \inR^n}. Consequently, we discuss the cases when this identity acts as Jordan derivation and 2- Jordan multiplier on prime rings