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Strong commutativity preserving generalized derivations on multilinear polynomials

Abstract 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.