Commun. Korean Math. Soc. 2017; 32(2): 233-260
Online first article September 28, 2016 Printed April 30, 2017
https://doi.org/10.4134/CKMS.c160069
Copyright © The Korean Mathematical Society.
Andreas Reinhart
Karl-Franzens-Universit\"at
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 ${\rm Int}(R)$.
Keywords: monadically Krull, integer-valued polynomial, divisor-class group
MSC numbers: 13A15, 13F05, 13F15, 20M12
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