When is a Nakayama closure semiprime?
type: Item
instance of: scholarly article
title: When is a Nakayama closure semiprime?
zbMATH Open document ID: 1326.13001
DOI: 10.1216/JCA-2014-6-3-439
author: Janet Vassilev
published in: Journal of Commutative Algebra
publication date: 2014-11-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1416233326
review text: If \(R\) is a commutative ring, a closure operation \(c\) on \(R\) is a function that associates with each ideal \(I\) an ideal \(c(I)=I^{c}\) containing \(I\) such that the inclusion of ideals is preserved and \(c(c(I))=c(I)\). For a local ring \(R\) with maximal ideal \(\mathfrak{m}\), the author studies the connections between semiprime operations (i.e., operations where \((I^c J^c)^c=(IJ)^c\) for every ideals \(I\) and \(J\)), Nakayama operations (i.e., operations for which \(I \subseteq J \subseteq (I+ \mathfrak{m}J)^c\) implies that \(I^c=J^c\)) and Krull operations (\((0)^c=\bigcap_{n \geq 1}(I^n)^c\) for all ideals \(I\)). When \(R\) is a discrete valuation ring, it is proved that a closure operation is Nakayama if and only if it is semiprime and Krull. Further, the author asks whether a Krull semiprime operation is Nakayama in every one dimensional domain. The paper also contains examples of Nakayama operations that are not semiprime and examples of semiprime operations that are not Nakayama.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 6370371
zbMATH Keywords: closure operation
zbMATH Keywords: semiprime operation
zbMATH Keywords: Nakayama operation
reviewed by: Catalin Ciuperca
MaRDI profile type: Publication
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OpenAlex ID: W2022798180
description: scientific article; zbMATH DE number 6370371