Residue class domains of the ring of convergent sequences and of \(C^{\infty}([0,1],{\mathbb{R}})\)
type: Item
instance of: scholarly article
title: Residue class domains of the ring of convergent sequences and of \(C^{\infty}([0,1],{\mathbb{R}})\)
zbMATH Open document ID: 0731.54014
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let \(N^*\) denote the one-point compactification of the integers N, and \(C(N^*,R)\) denote the ring of continuous mappings from \(N^*\) into the real numbers R, that is to say, \(C(N^*,R)\) is the ring of real convergent sequences. The author studies the residue class domains of \(C(N^*,R)\) and shows that there are precisely ten up to isomorphism. In this work, significant roles are played by non-principal ultrafilters on N and the Ax-Kochen theory of valuation fields. The author uses the following observation to extend these results: if X is a non-discrete metric space, then \(N^*\) can be C-embedded in X, and every residue class domain of \(C(N^*,R)\) is a residue class domain of C(X,R). Furthermore, if X is a topological space with a compact non-discrete subset, then C(X,R) includes four of the residue class rings of \(C(N^*,R)\). More results of this kind are proved for other classes of spaces X. The author then turns to the ring \(C^{\infty}(X,R)\) of \(C^{\infty}\)-functions on a closed interval X. Using rather intricate constructions, he classifies up to isomorphism twenty residue class domains of \(C^{\infty}(X,R)\), which are derived from those of \(C(N^*,R)\), and shows by means of an example that there are still more of them.
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 54C45
zbMATH DE Number: 4209057
zbMATH Keywords: valued fields
zbMATH Keywords: prime ideals
zbMATH Keywords: differential rings
zbMATH Keywords: residue class domain
author: James John Moloney
MaRDI profile type: Publication
full work available at URL: pjm.1990.143.79
OpenAlex ID: W2051305560
DOI: 10.2140/PJM.1990.143.79
description: scientific article; zbMATH DE number 4209057