On the divisor-class group of monadic submonoids of rings of integer-valued polynomials

TL;DR

This paper explores the divisor-class groups of monadic submonoids within rings of integer-valued polynomials over factorial domains, revealing their structure and implications for the arithmetic properties of these rings.

Contribution

It provides explicit descriptions of divisor-class groups of monadic submonoids of integer-valued polynomial rings and links their structure to the arithmetic of the larger ring.

Findings

Divisor-class groups of monadic submonoids are explicitly described in several cases.

Monadic submonoids are Krull monoids, facilitating their arithmetic analysis.

The structure of these submonoids explains properties like elasticity and tame degree in integer-valued polynomial rings.

Abstract

Let $R$ be a factorial domain. In this work we investigate the connections between the arithmetic of ${\rm Int}(R)$ (i.e., the ring of integer-valued polynomials over $R$) and its monadic submonoids (i.e., monoids of the form $\{g\in {\rm Int}(R)\mid g\mid_{{\rm Int}(R)} f^k$ for some $k\in\mathbb{N}_0\}$ for some nonzero $f\in {\rm Int}(R)$). Since every monadic submonoid of ${\rm Int}(R)$ is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between ${\rm Int}(R)$ and its monadic submonoids. If $R=\mathbb{Z}$ or more generally if $R$ has sufficiently many "nice" atoms, then we prove that the infinitude of the elasticity and the tame degree of ${\rm Int}(R)$ can be explained by using the structure of monadic submonoids of ${\rm Int}(R)$.