Formally integrally closed domains and the rings \(R((X))\) and \(R\{\{X\}\}\)
type: Item
instance of: scholarly article
title: Formally integrally closed domains and the rings \(R((X))\) and \(R\{\{X\}\}\)
zbMATH Open document ID: 0905.13007
published in: Journal of Algebra
publication date: 1998-05-25T00:00:00Z
review text: Let \(R\) be an integral domain. Denote by \(F(R)\) the set of nonzero fractional ideals of \(R\) and by \(F^{*} (R)\) the finitely generated members of \(F(R)\). If \( I \in F(R)\), \(I_{v}: = (I^{-1})^{-1}\) and \(I_{t} : = \bigcup J_{v}\), where \(J\) runs over the set \(\{ J \in F^{*} (R)| J \subseteq I \}\). Moreover, for \(f \in RX\), \(A_{f}\) stands for the ideal of \(R\) generated by the coefficients of \(f\). The ring \(R\) is said to be formally integrally closed if \((A_{fg})_{t} = (A_{f} A_{g})_{t}\) for all nonzero \(f,g \in RX\). In this paper, the authors study the rings \(R((X))= RX_{N}\) and \(R((X)) = RX_{N_{t}}\), where \(N = \{f \in RX \mid A_{f} = R \}\) and \(N_{t} = \{f \in RX \mid (A_{f})_{t} = R \}\), as a natural extension of the well-known properties of \(R[X]_{N}\) and \(R[X]_{N_{v}}\). They essentially prove the equivalence of the following statements: (i) \(R\) is a Krull domain (resp., \(R\) is a Dedekind domain). (ii) \(R((X))\) is a Krull domain (resp., \(R((X))\) is a Dedekind domain). (iii) \(R((X))\) is a Euclidean domain (resp., \(R((X))\) is a Euclidean domain). (iv) Every (principal) ideal of \(R((X))\) is extended from \(R\) (resp., every (principal) ideal of \(R((X))\) is extended from \(R\). (v) \(R\) is formally integrally closed and every prime ideal of \(R((X))\) is extended from \(R\).
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13F07
zbMATH DE Number: 1139782
zbMATH Keywords: formally integrally closed domains
zbMATH Keywords: Krull domains
zbMATH Keywords: PID
zbMATH Keywords: Euclidean domains
zbMATH Keywords: power series ring
author: D. D. Anderson
author: B. G. Kang
reviewed by: Carlos Galindo
MaRDI profile type: Publication
full work available at URL: jabr.1997.7262
OpenAlex ID: W1991230713
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DOI: 10.1006/JABR.1997.7262
description: scientific article; zbMATH DE number 1139782