Derivations on multilinear polynomials in semiprime rings

Let R be a 2-torsion free semiprime K-algebra with unity, d a non- zero derivation of R and f(x(1)....,x(u)) a non-zero multilinear polynomial over K. Suppose that, for every r(1),..,r(n) is an element of R [d(f(r(1),..,r(n))), f (r(1)...,r(n))] is zero or invertible in R. Then either R is a division ring, or f (x(1),..,x(n)) is a central polynomial for R, or [f (u(1),..,u(n)), u]d(U) = 0, for every u(1),..,u(n), u is an element of U, the left Utumi quotient ring of R, that is there exists a central idempotent e of U such that d vanishes identically on cU and f(x(1),.., x(n)) is central in (1 - e)U. Moreover the last conclusion holds that if and only if[d(f (r(1)...,r(n))), f (r(1),..,r(n))] = 0, for every r(1),..,r(n) is an element of R. This paper continues a line of investigation in the literature concerning the relationship between the structure of an associative ring R anti the behaviour of some derivation defined on R ([15], [4]).