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Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C (= Z(U)) the extended centroid of R. Let 0 ≠ α £ R and f(xi, ..., xn) a multilinear polynomial over C which is noncentral valued on R. Suppose that G and H are two nonzero generalized derivations of R such that a(H(f (x))f (x) - f (x)G(f (x))) £ C for all x = (xi, ... , Xn) £ Rn. In this paper, we prove that one of the following holds: (1)f (xi,...,xn)2 is central valued on R and there exist b,p,q £ U such that H(x) = px + xb for all x £ R, G(x) = bx + xq for all x £ R with a(p - q) £ C; (2)there exist p,q £ U such that H(x) = px + xq for all x £ R, G(x) = qx for all x £ R with ap = 0; (3)f(xi ,..., xn)2 is central valued on R and there exist q £ U, A £ C and an outer derivation g of U such that H(x) = xq + Ax - g(x) for all x £ R, G(x) = qx + g(x) for all x £ R, with a £ C; (4)R satisfies 54 and there exist b,p £ U such that H(x) = px + xb for all x £ R, G(x) = bx + xp for all x £ R. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with center Z(R). A mapping F: R→ R is called a generalized derivation if there exists a derivation d : R→R such that F(xy) = F(x)y+xd(y) holds for all x,yϵR. In the present paper, our main object is to study the situations: (1) F(xy)-F(x)F (y)ϵ Z(R), (2) F(xy)+F(x)F(y)ϵ Z(R), (3) F(xy)-F(y)F(x) ϵ Z(R), (4) F(xy)+F(y)F(x) ϵ Z(R); for all x,y in some suitable subset of R. [ABSTRACT FROM AUTHOR]