On generalized derivations satisfying certain identities

Let R be a prime ring with char R ? 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities. © 2011 Springer Science+Business Media, Inc

Posner’s second theorem and some related annihilating conditions on lie ideals

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two non-zero generalized derivations of R. If [F(u), u]G(u) = 0 for all u ? L, then one of the following holds: (a) there exists ? ? C such that F(x) = ?x, for all x ? R; (b) R ? M2 (F), the ring of 2 × 2 matrices over a field F, and there exist a ? U and ? ? C such that F(x) = ax + xa + ?x, for all x ? R. © 2018, University of Nis. All rights reserved

Centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x1,.., xn) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r1,.., rn): ri ? R} be the set of all evaluations of f(x1,.., xn) in R, while A = {[G (f(r1,.., rn)), f(r1,.., rn)]: ri ? R}, and let CR(A) be the centralizer of A in R; i.e., CR(A) = {a ? R: [a, x] = 0, ?x ? A }. We prove that if A ? (0), then CR(A) = Z(R). © 2017, Pleiades Publishing, Ltd

Generalized Skew Derivations with Invertible Values on Multilinear Polynomials

Let R be a prime ring, f(X 1,..., X n) a multilinear polynomial which is not central-valued on R, and G a nonzero generalized skew derivation of R. Suppose that G(f(x 1,..., x n)) is zero or invertible for all x 1,..., x n ? R. Then it is proved that R is either a division ring or the ring of all 2 × 2 matrices over a division ring. This result simultaneously generalizes a number of results in the literature. © 2012 Copyright Taylor and Francis Group, LLC