The residue fields of a zero-dimensional ring
type: Item
instance of: scholarly article
title: The residue fields of a zero-dimensional ring
zbMATH Open document ID: 0932.13005
DOI: 10.1016/S0022-4049(97)00044-3
author: Roger Wiegand
author: William J. Heinzer
author: David C. Lantz
published in: Journal of Pure and Applied Algebra
publication date: 2000-03-08T00:00:00Z
review text: Let \({\mathcal K} = \{K_{\alpha}\}_{\alpha \in A}\) denote an indexed family of fields. In J. Pure Appl. Algebra 82, No. 2, 131-153 (1992; Zbl 0782.13014), \textit{R. Gilmer} and \textit{G. Heinzer} considered the question under what conditions there exists a zero-dimensional commutative ring \(R\) such that \(\mathcal K\) is the family of residue fields \(\{R/M_{\alpha}\}_{\alpha \in A}\) up to isomorphism? In the case \(\mathcal K\) is the family of the residue fields of a zero-dimensional commutative ring \(R\) the associated bijection from the index set \(A\) to the spectrum of \(R\) (endowed with the Zariski topology) gives \(A\) the topology of a Boolean space. The paper under review is concerned with the following problem: Given a field \(F,\) a Boolean space \(X\) and a family \(\{K_x\}_{x \in X}\) of extension fields of \(F\). Under which circumstances does there exist a zero-dimensional \(F\)-algebra \(R\) such that \(\mathcal K\) is up to \(F\)-isomorphism the family of residue fields of \(R\) such that the associated bijection from \(X\) to \(\text{Spec} (R)\) is a homomorphism? The authors consider the necessary condition: For a given \(x \in X\) and any finite field extension \(F \subseteq E \subseteq K_x,\) there exists a neighborhood \(V\) of \(x\) and, for each \(y \in V,\) an \(F\)-embedding of \(E\) into \(K_x.\) Then they prove several partial converses of this result, under a hypothesis which allows the ``straightening'' of the \(F\)-embeddings to make them compatible. The authors give particular attention to the case where \(X\) has only one accumulation point and where \(X\) is countable. Moreover, several examples are provided.
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 12F99
zbMATH DE Number: 1308189
zbMATH Keywords: residue fields
zbMATH Keywords: zero-dimensional ring
zbMATH Keywords: Boolean space
zbMATH Keywords: field extension
reviewed by: Peter Schenzel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308189