Power series rings over strong Mori domains.
type: Item
instance of: scholarly article
title: Power series rings over strong Mori domains.
zbMATH Open document ID: 1043.13004
author: Mi Hee Park
published in: Journal of Algebra
publication date: 2004-01-14T00:00:00Z
review text: Let \(R\) be an integral domain with quotient field \(K\). An ideal \(J\) of \(R\) is called Glaz-Vasconcelos ideal, denoted \(J\in GV(R)\), if \(J\) is finitely generated and \(J^{-1}=R\). For a fractional ideal \(I\) of \(R\), the \(w\)-closure of \(I\) is defined by \(I_w=\{x\in K;\, xJ\subseteq I\) for some \(J\in GV(R)\}\). \(I\) is called a \(w\)-ideal if \(I=I_w\). \(R\) is called a strong Mori domain if \(R\) satisfies the ascending chain condition on integral \(w\)-ideals. On the other hand, if \(\{X_{\lambda}; \lambda\in \Lambda\}\) is an arbitrary set of indeterminates over \(R\), the ring \(R[[X_{\lambda}; \lambda\in\Lambda]]_1\) is defined to be the union of the ascending net of the rings \(R[[X_{\lambda}; \lambda\in F]]\), where \(F\) runs over all finite subsets of \(\Lambda\). In this paper, the author shows: (1) if \(R\) is a noetherian domain, then \(R[[X_{\lambda}; \lambda\in\Lambda]]_1\) is a strong Mori domain; (2) if \(R\) is a strong Mori domain, then \(R[[X_{\lambda}; \lambda\in\Lambda]]_1\) is a Mori domain; but (3) there exists a strong Mori domain \(R\) such that \(RX\) is not a strong Mori domain.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 2026933
zbMATH Keywords: integral domain
zbMATH Keywords: strong Mori domain
zbMATH Keywords: power series rings
zbMATH Keywords: Glaz-Vasconcelos ideal
zbMATH Keywords: \(w\)-ideal
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.07.009
OpenAlex ID: W2023718222
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DOI: 10.1016/J.JALGEBRA.2003.07.009
description: scientific article; zbMATH DE number 2026933