Generalized derivations acting on multilinear polynomials in prime rings

summary:Let R be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,… ,xₙ) a multilinear polynomial over C which is not central valued on R. If F(f(r))G(f(r))=H(f(r)²) for all r=(r₁,… ,rₙ) ∈ Iⁿ, then one of the following conditions holds: {(1)} there exist a∈ C and b∈ U such that $F(x)=ax$, $G(x)=xb$ and $H(x)=xab$ for all x∈ R; {(2)} there exist a, b∈ U such that $F(x)=xa$, $G(x)=bx$ and $H(x)=abx$ for all x∈ R, with ab∈ C; {(3)} there exist b∈ C and a∈ U such that $F(x)=ax$, $G(x)=bx$ and $H(x)=abx$ for all x∈ R; {(4)} f(x₁,… ,xₙ)² is central valued on R and one of the following conditions holds: {(a)} there exist a,b,p,p'∈ U such that $F(x)=ax$, $G(x)=xb$ and $H(x)=px+xp'$ for all x∈ R, with $ab=p+p'$; {(b)} there exist a,b,p,p'∈ U such that $F(x)=xa$, $G(x)=bx$ and $H(x)=px+xp'$ for all x∈ R, with p+p'=ab∈ C.