Finiteness properties of differential polynomials

Let R be a prime ring with extended centroid C and let be a reduced differential polynomial with coefficients in Q, the symmetric Martindale quotient ring of R, and with zero constant term. Let and Bϕ={ϕ(xij)|xij∈R}. We prove that the finiteness of Aϕ and the finite-dimensionality of the C-span of Aϕ are equivalent to that of Bϕ and that of the C-span of Bϕ, respectively. Hence some questions on differential polynomials are reduced to those on ordinary generalized polynomials.Let δ and d be two derivations of R,L a Lie ideal of R and ρ a right ideal of R. As applications of our theorems, we obtain the necessary and sufficiency conditions for the finiteness of d(ρ),d(L) and δd(L) and for the finite-dimensionality of the C-spans of d(ρ),d(L) and δd(L).