On \(t\)-reductions of ideals in pullbacks
type: Item
instance of: scholarly article
title: On \(t\)-reductions of ideals in pullbacks
zbMATH Open document ID: 1360.13055
published in: The New York Journal of Mathematics
publication date: 2016-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1607.06705
full work available at URL: 22-41.html
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 6636848
zbMATH Keywords: reduction
zbMATH Keywords: basic ideal
zbMATH Keywords: basic ring
zbMATH Keywords: \(t\)-operation
zbMATH Keywords: \(t\)-reduction
zbMATH Keywords: \(t\)-basic ideal
zbMATH Keywords: pseudo-valuation domain
zbMATH Keywords: PvMD
zbMATH Keywords: \(v\)-domain
zbMATH Keywords: integrally closed domain
author: S. Kabbaj
author: A. Kadri
author: A. Mimouni
reviewed by: Malgorzata Marciniak
MaRDI profile type: Publication
arXiv ID: 1607.06705
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review text: Reductions considered in the article are generalizations of the concept introduced by \textit{D. G. Northcott} and \textit{D. Rees} [Proc. Camb. Philos. Soc. 50, 145--158 (1954; Zbl 0057.02601)]. They are related but distinct from \textit{N. Epstein}'s \(c\)-reductions [J. Algebra 323, No. 8, 2209--2225 (2010; Zbl 1243.13002)].NEWLINENEWLINELet \(R\) be a domain and \(I\) a nonzero fractional ideal of \(R\), then the \(v-\), \(t-\), and \(w-\) closures are defined respectively as \(I_v=\left(I^{-1}\right)^{-1}\), \(I_t=\bigcup J_{v}\), where \(J_v\) ranges over the set of finitely generated subideals of \(I\), and \(I_w=\bigcap IR_M\), where \(M\) ranges over the set of maximal \(t-\)ideals of \(R\). An ideal \(J\) is a \(*-\)reduction of \(I\) if \(\left(JI^n\right)^{*}=\left(I^{n+1}\right)^{*}\). An ideal that has no \(*-\)reductions but itself is called \(*-\)basic and a ring has \(*-\)basic ideal property if all its finitely generated ideals are \(*-\)basic, with \(v\), \(t\) or \(w\) substituted for \(*\). The notions of reduction and \(t\)-reduction coincide for Prüfer domains and the authors generalize this result by find necessary and sufficient conditions for the reduction and \(t\)-reduction to coincide in pseudo-valuation domains. In addition the article contains an example when the reduction and \(t\)-reduction of an ideal are distinct.NEWLINENEWLINEThe main result of the article states that if \(R\) is a domain then the notions of \(t\)-basic and \(v\)-basic ideals coincide, which is a generalization of results by \textit{J. H. Hays} [Trans. Am. Math. Soc. 177, 51--63 (1973; Zbl 0266.13001); Proc. Am. Math. Soc. 52, 81--84 (1975; Zbl 0345.13013)], and solves an open problem stated in [\textit{E. Houston} et al., ``\(*\)-reductions of ideals and Prüfer \(v\)-multiplication domains'', Preprint, \url{arXiv:1602.07035}].NEWLINENEWLINEFollowing the pullback diagram admitted by pseudo-valuation domains, the authors consider the following diagram denoted as \(…
description: scientific article; zbMATH DE number 6636848