STRONG COMMUTATIVITY PRESERVING GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS

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 is an element of 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 (x(1), . . . , x(n)) on R. In all cases, if R does not satisfy s4(x(1), . . . , x(1)), the standard polynomial identity on 4 non-commuting variables, then there exist s, c is an element of U such that F (x) = xs, G(x) = cx, for all x is an element of R, and sc = 1(C) (the unit of C). We also study the semiprime case.