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 is an element of K[X] vertical bar f(a) is an element of D for all a is an element of 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 a is an automorphism of K, one may consider the set Int(D, sigma) = { f is an element of K[X, sigma] vertical bar f(a) is an element of D for all a is an element of D} , where K[X, sigma] 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, sigma) 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, sigma, delta] with a suitable evaluation, where delta is a sigma-derivation. Moreover we prove, for example, that if sigma is of finite order and D is a Dedekind domain with finite residue fields such that Int(D, sigma) is a ring, then Int(D, sigma) is non-Noetherian.