Weakly factorial property of a generalized Rees ring \(D[X,d/X]\)
type: Item
instance of: scholarly article
title: Weakly factorial property of a generalized Rees ring \(D[X,d/X]\)
zbMATH Open document ID: 1404.13006
DOI: 10.1216/RMJ-2018-48-7-2175
author: Gyu Whan Chang
published in: Rocky Mountain Journal of Mathematics
publication date: 2019-01-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1544756806
review text: This article investigates the weakly factorial property of a generalized Rees ring \(R:=D[X,d/X]\) where \(D\) is an integral domain, \(d\in D\), and \(X\) is an indeterminate over \(D\). An integral domain \(D\) is said to be a weakly factorial domain (WFD) if each nonzero, nonunit of \(D\) can be written as a finite product of primary elements of \(D\), where \(a\in D\) is primary if \(aD\) is a primary ideal. This article looks to expand on some similar results in the literature about the ring \(S=D[tI, t^{-1}]\) where \(t\) is an indeterminate and \(I\) is an ideal such as: (1) if \(I\) is finitely generated then \(S\) is a UFD if and only if \(D\) is a UFD and \(t^{-1}\) is a prime element of \(S\) [\textit{D. Whitman}, Math. Jap. 17, 13--14 (1972; Zbl 0255.13015)], (2) \(S\) is a GCD-domain if and only if \(D\) is a GCD-domain and \(t^{-1}\) is a prime element of \(S\) [\textit{J. Mott}, Math. Jap. 20, 85--87 (1975; Zbl 0321.13001)] , (3) \(S\) is a Prüfer \(\nu\)-multiplication domain (P\(\nu\)MD) if and only if \(S\) is a (P\(\nu\)MD), \(t^{-1}\) is prime in \(S\) and \(\cap_{n=1}^{\infty} I^n = (0)\) [\textit{H. Kim} et al., Bull. Korean Math. Soc. 41, No. 3, 473--482 (2004; Zbl 1074.13002)]. We note that the situation focused on in this article is achieved when \(I=dD\) and \(X=t^{-1}\). The author is able to prove several nice results including, but not limited to the following: (1) \(R\) is a WFD if and only if \(R\) is a weakly factorial GCD-domain if and only if \(D\) is a weakly factorial GCD domain and \(d=0\), \(d \in U(D)\), or \(d\) is prime in \(D\), (2) \(R\) is a ring of Krull type if and only if \(D\) is of Krull type, and (3) If \(D\) is a weakly factorial GCD-domain, \(p\) is prime in \(D\), and \(n \geq 2\) is an integer, then \(D[X, p^n/X]\) is an almost weakly factorial domain with Cl\((D[X, p^n/X])=\mathbb{Z}_n\). A domain is said to be of Krull type if there is a set \(\{V_\alpha\}\) of valuation overrings of \(D\) such that (1) each \(…
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 6999259
zbMATH Keywords: weakly factorial domain
zbMATH Keywords: GCD-domain
zbMATH Keywords: generalized Rees ring \(D[X,d/x]\)
reviewed by: Christopher P. Mooney
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6999259