Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(1): 149-171

Online first article July 17, 2020      Printed January 1, 2021

https://doi.org/10.4134/JKMS.j200010

Copyright © The Korean Mathematical Society.

Every abelian group is the class group of a ring of Krull type

Gyu Whan Chang

Incheon National University

Abstract

Let $Cl(A)$ denote the class group of an arbitrary integral domain $A$ introduced by Bouvier in 1982. Then $Cl(A)$ is the ideal class (resp., divisor class) group of $A$ if $A$ is a Dedekind or a Pr\"ufer (resp., Krull) domain. Let $G$ be an abelian group. In this paper, we show that there is a ring of Krull type $D$ such that $Cl(D) = G$ but $D$ is not a Krull domain. We then use this ring to construct a Pr\"ufer ring of Krull type $E$ such that $Cl(E) = G$ but $E$ is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

Keywords: Krull domain, P$v$MD, ring of Krull type, Pr\"ufer domain, class group, polynomial ring

MSC numbers: 13A15, 13F05

Supported by: This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867).