A note on integer-valued skew polynomials.

Given an integral domain D with quotient field K , the study of the ring of integer-valued polynomials Int (D) = { f ∈ K [ X ] | f (a) ∈ D for all a ∈ D } has attracted a lot of attention over the past decades. Recently, Werner has extended this study to the situation of skew polynomials. To be more precise, if σ is an automorphism of K , one may consider the set Int (D , σ) = { f ∈ K [ X , σ ] | f (a) ∈ D for all a ∈ D } , where K [ X , σ ] is the skew polynomial ring and f (a) is a "suitable" evaluation of f at a. For example, he gave sufficient conditions for Int (D , σ) to be a ring and study some of its properties. In this paper, we extend the study to the situation of the skew polynomial ring K [ X , σ , δ ] with a suitable evaluation, where δ is a σ -derivation. Moreover we prove, for example, that if σ is of finite order and D is a Dedekind domain with finite residue fields such that Int (D , σ) is a ring, then Int (D , σ) is non-Noetherian. [ABSTRACT FROM AUTHOR]