A note on integer-valued skew polynomials

Abstract

Given an integral domain [Formula: see text] with quotient field [Formula: see text], the study of the ring of integer-valued polynomials [Formula: see text] 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 [Formula: see text] is an automorphism of [Formula: see text], one may consider the set [Formula: see text], where [Formula: see text] is the skew polynomial ring and [Formula: see text] is a “suitable” evaluation of [Formula: see text] at [Formula: see text]. For example, he gave sufficient conditions for [Formula: see text] 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 [Formula: see text] with a suitable evaluation, where [Formula: see text] is a [Formula: see text]-derivation. Moreover we prove, for example, that if [Formula: see text] is of finite order and [Formula: see text] is a Dedekind domain with finite residue fields such that [Formula: see text] is a ring, then [Formula: see text] is non-Noetherian.