Generalized derivations with multilinear polynomials in prime rings

摘要

Let R be a prime ring of characteristic different from two with Utumi quotient ring U and extended centroid C, f (x(1), ..., x(n)) be a multilinear polynomial over C, which is not central valued on R. Suppose that F, G and H are three generalized derivations on R. If F(f(r))G(f(r))-f(r)H(f(r)) = 0 for all r = (r(1), ...., r(n)) is an element of R-n, then one of the following holds: