Prime ideals in skew polynomial rings and quantized Weyl algebras

The concern of this paper is to investigate the structure of skew polynomial rings (Ore extensions) of the form T=R[θ; σ, δ] where σ and δ are both nontrivial, and in particular to analyze the prime ideals of T. The main focus is on the case that R is commutative noetherian. In this case, the prime ideals of T are classified, polynomial identities and Artin-Rees separation in prime factor rings are investigated, and cliques of prime ideals are studied. The second layer condition is proved, as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique. Futher, q-skew derivations on noncommutative coefficient rings are introduced, and some preliminary results on contractions of prime ideals of T are obtained in this setting. Finally, prime ideals in quantized Weyl algebras over fields are analyzed