Ring theory and pointfree topology.
type: Item
instance of: scholarly article
title: Ring theory and pointfree topology.
zbMATH Open document ID: 1040.06004
DOI: 10.1016/S0166-8641(03)00196-2
author: Bernhard Banaschewski
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: In this paper the author reviews the `pointfree' version of the sheaf representation theory of rings (using sheaves on locales rather than on classical topological spaces). The advantage of working with locales in this context is that one can eliminate the uses of the Boolean Prime Ideal Theorem, which are required in the classical case to ensure that the rings in question have enough prime ideals (equivalently, that the locales over which they are represented have enough points). As an example, he gives a pointfree (and hence choice-free) proof of an old result of the reviewer, that a ring is an exchange ring if and only if it is a Gelfand ring and its maximal spectrum is zero-dimensional. He also considers the sheaf representation of ordered rings, and shows that an \(l\)-ring is an \(f\)-ring if and only if it has a sheaf representation by a totally ordered ring; once again, the `pointed' version of this result is in the reviewer's book [Stone spaces. Cambridge: Cambridge Univ. Press (1982; Zbl 0499.54001)].
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 2056840
zbMATH Keywords: sheaf representation theory
zbMATH Keywords: sheaves on locales
zbMATH Keywords: exchange ring
zbMATH Keywords: Gelfand ring
zbMATH Keywords: ordered rings
reviewed by: Peter Johnstone
MaRDI profile type: Publication
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full work available at URL: s0166-8641(03)00196-2
OpenAlex ID: W1967639792
description: scientific article; zbMATH DE number 2056840