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

TL;DR

This paper characterizes b-generalized skew derivations on prime rings that satisfy a specific identity involving multilinear polynomials, revealing their structure and conditions under which they act as Jordan derivations or 2-Jordan multipliers.

Contribution

It provides a complete description of b-generalized skew derivations satisfying a key identity on prime rings, extending understanding of their behavior in algebraic structures.

Findings

Characterization of b-generalized skew derivations satisfying the identity.

Conditions under which the identity acts as Jordan derivation.

Cases where the identity functions as 2-Jordan multiplier.

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 \phi be 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 {{\phi}({\zeta}) | {\zeta} = ({\zeta}1 . . . , {\zeta}n) \inR^n}. Consequently, we discuss the cases when this identity acts as Jordan derivation and 2- Jordan multiplier on prime rings