Primary ideals with finitely generated radical in a commutative ring
type: Item
instance of: scholarly article
title: Primary ideals with finitely generated radical in a commutative ring
zbMATH Open document ID: 0794.13002
DOI: 10.1007/BF02599309
author: Robert Gilmer
author: William J. Heinzer
published in: Manuscripta Mathematica
publication date: 1994-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/155804
review text: This paper consider the question of when a finitely generated prime ideal \(P\) has every \(P\)-primary ideal finitely generated. The authors call a prime ideal with this property primarily finite and call a ring primarily finite if each of its finitely generated prime ideals is primarily finite. An example is given of a strongly Laskerian ring with only two prime ideals \(P \subset M\) and with \(P\) finitely generated and \(P\) having only one primary ideal different from \(P\) and with that primary ideal not finitely generated. Thus a finitely generated prime ideal need not be primarily finite. It is shown that while a polynomial ring over a primarily finite ring need not be primarily finite, a polynomial ring in one indeterminate over a Prüfer domain or in any number of indeterminates over a one-dimensional Prüfer domain or any Noetherian ring is primarily finite. The paper contains a number of other interesting related results.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 492772
zbMATH Keywords: primary ideal
zbMATH Keywords: finitely generated prime ideal
zbMATH Keywords: polynomial ring over a primarily finite ring
reviewed by: D. D. Anderson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492772