Centralizing b-generalized derivations on multilinear polynomials

Abstract

Let R be a prime ring of characteristic different from 2 and F a b-generalized derivation on R. Let U be Utumi quotient ring of R with extended centroid C and f (x 1 ,. .. , x n) be a multilinear polynomial over C which is not central valued on R. Suppose that d is a non zero derivation on R such that d([F(f (r)), f (r)]) ∈ C for all r = (r 1 ,. .. , r n) ∈ R n ; then one of the following holds: (1) there exist a ∈ U, λ ∈ C such that F(x) = ax + λx + xa for all x ∈ R and f (x 1 ,. .. , x n) 2 is central valued on R, (2) there exists λ ∈ C such that F(x) = λx for all x ∈ R.

b-Generalized Skew Derivations on Multilinear Polynomials

“…More precisely, they describe the structure of additive mapping satisfying the identity F (f (r))f (r) − f (r)G(f (r)) = 0 for all r ∈ R n , where f is a multilinear polynomial and F , G are two generalized derivations on prime ring R. In 2018, Tiwari [19] studied the commuting generalized derivations on prime ring, which is generalization of the work of Argaç and De Filippis [16]. The generalization of Posner's theorem for generalized derivation on multilinear polynomial in [26] (where further generalization can be found in [1,2,20,21]) is given below.…”