Prime ideals in ultraproducts of commutative rings
type: Item
instance of: scholarly article
title: Prime ideals in ultraproducts of commutative rings
zbMATH Open document ID: 1097.13032
author: Bruce Olberding
author: Jay Shapiro
published in: Journal of Algebra
publication date: 2005-04-18T00:00:00Z
review text: Context: \(I\) is an infinite set and \(\mathcal U\) its ultrafilter. \(\{R_i\}_{i\in I}\) is a set of commutative rings; \(\prod_{\mathcal U}R_i\) denotes the ultraproduct mod \(\mathcal U\). The paper gives a number of consequences of a well-known Theorem of Łoś': If \(\phi\) is a sentence in the language of commutative rings, then \(\prod_{\mathcal U}R_i\) satisfies \(\phi\) if and only if \(R_i\) satisfies \(\phi\) for \(\mathcal U\)-many \(i\) (i.e. a property \(\mathcal S\) holds for \(\mathcal U\)-many \(i\), if the set of all \(i\) such that \(R_i\) satisfies \(\mathcal S\) is a member of \(\mathcal U\)). The goal of the paper is to describe classes of prime ideals of ultraproducts of commutative rings (such as Noetherian rings, Krull domains, QR-domains and finite character rings). If \(R\) is a commutative ring and \(V\) a non-empty set of prime ideals of \(R\), then denote \(\mathbf J(V)=\bigcap_{P\in V}P\). If \(\mathcal L\) is a sublattice of the lattice of all subsets of \(\mathcal P\subseteq \text{Spec}(R)\), for a proper filter \(\mathcal F\) define \((\mathcal F)=\bigcup_{V\in\mathcal F}\mathbf J(V)\) and if \(Q\) is a prime ideal of \(R\), then denote \(\mathcal F(Q)=\{ V\in\mathcal L:\mathbf J(V)\subseteq Q\}\). Then, several results establish equivalent conditions when the mappings \(\mathcal F\mapsto (\mathcal F)\) and \(Q\mapsto \mathcal F(Q)\) form bijective correspondences between filters and ideals (one such equivalent condition is that \(\mathcal L\) be saturated).
reviewed by: Radoslav M. Dimitrić
Mathematics Subject Classification ID: 13L05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 2157759
zbMATH Keywords: (weakly
zbMATH Keywords: strongly) saturated lattice
zbMATH Keywords: basis lattice
zbMATH Keywords: Prüfer domain
zbMATH Keywords: ultra-height
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.11.004
OpenAlex ID: W1999790326
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DOI: 10.1016/J.JALGEBRA.2004.11.004
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description: scientific article; zbMATH DE number 2157759