Semistar ascending chain conditions over power series rings
type: Item
instance of: scholarly article
title: Semistar ascending chain conditions over power series rings
zbMATH Open document ID: 1477.13044
author: Gmiza Wafa
author: Sana Hizem
published in: Ricerche di Matematica
publication date: 2021-12-03T00:00:00Z
review text: Let \(D\) be an integral domain, \(\star\) a semistar operation on \(D\) and \(\star_f\) the semistar operation of finite type associated to \(\star.\) \textit{S. El Baghdadi} et al. [J. Pure Appl. Algebra 193, No. 1--3, 27--60 (2004; Zbl 1081.13003)] introduced the notion of semistar Noetherian domains. In the second section of this paper, the authors associate to a semistar operation \(\star\) on \(D\), a semistar operation \(\ast\) on \(DX\). They show that if \(D\) is a semistar Noetherian domain and if \(D\) satisfies the \(\star\)-finite property, that is each nonzero element \(x\) of \(D\) belongs to only finitely many quasi-\(\star\)-maximal ideal of \(D\), then \(DX\) is a \(\ast\)-Noetherian domain. In the third section, the authors are interested in domains satisfying the ascending chain condition on radical quasi semistar ideals. This notion is defined by \textit{P. Sahandi} [Rocky Mt. J. Math. 40, No. 3, 1039--1049 (2010; Zbl 1200.13008)]. They give sufficient conditions on \(D\) and the semistar operation on \(D\) for \(DX\) to satisfy the ascending chain condition on radical quasi-\(\ast\)-ideals.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13E99
zbMATH DE Number: 7438461
zbMATH Keywords: semistar operation
zbMATH Keywords: ascending chain condition
zbMATH Keywords: formal power series ring
MaRDI profile type: Publication
full work available at URL: s11587-019-00474-8
OpenAlex ID: W2996290642
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DOI: 10.1007/S11587-019-00474-8
description: scientific article; zbMATH DE number 7438461