Ore extensions and Poisson algebras

For a derivation d of a commutative Noetherian complex algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;d] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=d(a) for all a in A. Several illustrative examples are discussed including some where A is known to be d-simple.