Prime ideals and finiteness conditions for Gabriel topologies over commutative rings
type: Item
instance of: scholarly article
title: Prime ideals and finiteness conditions for Gabriel topologies over commutative rings
zbMATH Open document ID: 0772.13007
author: Morris Orzech
author: Margaret Beattie
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-01-17T00:00:00Z
review text: The authors study the Gabriel topologies on rings with restriction either on ideals from that topologies or on chains of ideals from them. If \(P\) is a set of prime ideals of \(R\), then by \({\mathfrak C}(P)\) is denoted the Gabriel topology on \(R\) consisting from those ideals which are not contained in \(p\), where \(p\in P\). The Gabriel topologies of the form \({\mathfrak C}(P)\) where \(P\subseteq\text{Spec}(R)\) are called primal. A Gabriel topology is called of finite type (principal) if it has a base consisting from finitely generated (principal) ideals. It is proved (theorem 1.1) that a Gabriel topology \({\mathfrak C}\) is primal \(\Leftrightarrow{\mathfrak C}\) is an intersection of topologies of finite type \(\Leftrightarrow{\mathfrak C}\) is an intersection of principal topologies. A Gabriel topology \({\mathfrak C}\) is of finite type if and only if it satisfies the following condition: If the union of a totally ordered chain of ideals is in \({\mathfrak C}\), then one of the ideals must be in \({\mathfrak C}\) (theorem 1.2). The paper contains in terms of the Gabriel topologies a generalization of a theorem of \textit{I. S. Cohen} which says that if every prime ideal of a commutative ring \(R\) is finitely generated, then \(R\) is noetherian. It is proved, for example, that if every prime ideal from a Gabriel topology \({\mathfrak C}\) is finitely generated, then every ideal from \({\mathfrak C}\) is finitely generated (corollary 1). Special kinds of primal topologies are studied in the paper and some constructions of primal topologies are given. Some examples are given which show that a number of classes of Gabriel topologies are different.
Mathematics Subject Classification ID: 13J99
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 54H13
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 97416
zbMATH Keywords: prime ideals
zbMATH Keywords: finiteness conditions
zbMATH Keywords: Gabriel topologies
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072739
OpenAlex ID: W2088277016
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DOI: 10.1216/RMJM/1181072739
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description: scientific article; zbMATH DE number 97416