Strong commutativity preserving generalized derivations on multilinear polynomials
Articles > Article · 2017
Authors: Nurcan Argaç, Giovanni Scudo
Quaestiones Mathematicae, International, WoS, 2017, vol 40, pp 1075-1094
DOI: 10.2989/16073606.2017.1348398
Summary
Let R be a non-commutative prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C and let F and G be generalized derivations of R such that F (x)G(y) − F (y)G(x) = [x, y], for all x, y ∈ S, where S is a subset of R. Here we will discuss the following cases:(a) S = [R, R];(b) S = L, where L is a non-central Lie ideal of R;(c) S = f (R), where f (R) is the set of all evaluations of a non-central multilinear polynomial f (x1, … , xn) on R.In all cases, if R does not satisfy s4(x1, … , x4), the standard polynomial identity on 4 non-commuting variables, then there exist s, c ∈ U such that F (x) = xs, G(x) = cx, for allx ∈ R, and sc = 1C (the unit of C).We also study the semiprime case.