The family of residue fields of a zero-dimensional commutative ring
type: Item
instance of: scholarly article
title: The family of residue fields of a zero-dimensional commutative ring
zbMATH Open document ID: 0782.13014
DOI: 10.1016/0022-4049(92)90117-X
author: Robert Gilmer
author: William J. Heinzer
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: Let \({\mathcal F}\) denote a family of fields. The theme of this paper is the realizability of \({\mathcal F}\): that is, under what conditions on \({\mathcal F}\) does there exist a commutative zero-dimensional ring \(R\) such that the family \({\mathcal F}(R)\) of residue fields of \(R\) is in bijection with \({\mathcal F}\)? For example, it is shown that if \({\mathcal F}\) includes a finite subset \(\{F_ 1,\ldots,F_ n\}\) such that every field in \({\mathcal F}\) contains an isomorphic copy of (at least) one of the \(F_ i\), then \({\mathcal F}\) is realizable. In the reverse direction, if \({\mathcal F}\) consists of finite fields, or if every field in \({\mathcal F}\) is a finite dimensional vector space over a fixed field \(K\), then \({\mathcal F}\) is realizable only if it includes a finite set of minimal members. However, this converse result fails in general: For every prime \(p\) there is a realizable set \({\mathcal F}=\{K_ i:0 \leq i<\infty\}\) with each \(K_ i\) of characteristic \(p\), \(K_ 0\) infinite, and all \(K_ i\) distinct, finite and minimal, for \(i \geq 1\).
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 12E99
zbMATH DE Number: 166140
zbMATH Keywords: family of residue fields of a commutative zero-dimensional ring
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: finite fields
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Q5563439
cites work: On strongly πregular rings and homomorphisms into them
cites work: The Gluing of Maximal Ideals-Spectrum of a Noetherian Ring-Going up and Going Down in Polynomial Rings
cites work: On the imbedding of a regular ring in a regular ring with identity
cites work: Q3270120
cites work: Some Countability Conditions on Commutative Ring Extensions
cites work: Jónsson \(\omega _ 0\)-generated algebraic field extensions
cites work: Products of Commutative Rings and Zero-Dimensionality
cites work: Zero-Dimensionality in Commutative Rings
cites work: Q3229455
cites work: Sur la structure des anneaux absoluments plats commutatifs
cites work: Q3943047
cites work: Globalization theorems for locally finitely generated modules
cites work: Modules over universal regular rings
description: scientific article; zbMATH DE number 166140