Journal of the
Korean Mathematical Society
JKMS

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

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J. Korean Math. Soc. 2023; 60(2): 407-464

Online first article February 20, 2023      Printed March 1, 2023

https://doi.org/10.4134/JKMS.j220271

Copyright © The Korean Mathematical Society.

Prime factorization of ideals in commutative rings, with a focus on Krull rings

Gyu Whan Chang, Jun Seok Oh

Incheon National University; Jeju National University

Abstract

Let $R$ be a commutative ring with identity. The structure theorem says that $R$ is a PIR (resp., UFR, general ZPI-ring, $\pi$-ring) if and only if $R$ is a finite direct product of PIDs (resp., UFDs, Dedekind domains, $\pi$-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations $v$ or $t$ as follows: An integral domain $R$ is a Krull domain if and only if every nonzero proper principal ideal of $R$ can be written as a finite $v$- or $t$-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation $u$ on $R$, so that $R$ is a general Krull ring if and only if every proper principal ideal of $R$ can be written as a finite $u$-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

Keywords: Krull domain, SPR, general Krull ring, $u$-operation, Nagata ring, Noetherian ring

MSC numbers: 13A15, 13B25, 13E05, 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).