Research ArticleNo Access

A note on integer-valued skew polynomials

Angelot Behajaina

Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France

E-mail Address: angelot.behajaina@universite-paris-saclay.fr

https://doi.org/10.1142/S0219498823501712Cited by:0 (Source: Crossref)

Abstract

Given an integral domain $D$ with quotient field $K$ , the study of the ring of integer-valued polynomials $Int (D) = {f \in K [X] | f (a) \in D for all a \in D}$ 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 $Int (D, σ) = {f \in K [X, σ] | f (a) \in D for all a \in D}$ , 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.

Communicated by André Leroy

Keywords:

AMSC: 16S36, 13F20

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