Aleph-nought-continuous regular rings
type: Item
instance of: scholarly article
title: Aleph-nought-continuous regular rings
zbMATH Open document ID: 0618.16012
DOI: 10.1016/0021-8693(87)90167-0
author: Pere Ara
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Although aleph-nought-continuous (von Neumann) regular rings do not possess the Type I, II, III structural decomposition theory available for continuous or self-injective regular rings, they nonetheless have a good structure theory (based on factor rings rather than summands). For right and left aleph-nought-continuous regular rings, this was initiated by \textit{D. Handelman} [Adv. Math., Suppl. Stud., Vol. 4, 171-196 (1979; Zbl 0511.46054)] and continued by him together with \textit{D. Higgs} and \textit{J. Lawrence} [J. Lond. Math. Soc., II. Ser. 21, 193-202 (1980; Zbl 0449.06013)] and \textit{J. Lawrence} and the reviewer [Mem. Am. Math. Soc. 234 (1980; Zbl 0435.16005)]. The reviewer later extended most of this structure theory to directly finite right aleph-nought-continuous regular rings [Pac. J. Math. 100, 105-122 (1982; Zbl 0453.16007)]. In the present paper, the author pushes on to structure theory for an arbitrary right aleph-nought-continuous regular ring \(R\). His key result (comparable to the directly finite/purely infinite decomposition for a continuous regular ring) is that the set \(I\) of those \(x\in R\) for which \(R\oplus xR\) embeds in \(R\) is a two-sided ideal such that \(R/I\) is directly finite and right aleph-nought-continuous. Using various lifting techniques, he is then able to remove the directly finite hypothesis from a number of earlier results. For example, if \(A\) and \(B\) are finitely generated projective right \(R\)-modules such that \(nA\cong nB\) (or \(nA\) embeds in \(nB\)) for some positive integer \(n\), then \(A\cong B\) (or \(A\) embeds in \(B\)). Two open problems from the reviewer's book [Von Neumann regular rings (1979; Zbl 0411.16007)] are answered (affirmatively) as well.
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 4002304
zbMATH Keywords: aleph-nought-continuous regular rings
zbMATH Keywords: directly finite right aleph-nought-continuous regular rings
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90167-0
OpenAlex ID: W1973872772
cites work: On \(\aleph _ 0\)-injective regular rings
cites work: Equivalence and perspectivity in Baer rings
cites work: Q4198895
cites work: Directly finite aleph-nought-continuous regular rings
cites work: Affine representations of Grothendieck groups and applications to Rickart 𝐶*-algebras and ℵ₀-continuous regular rings
cites work: Q4748637
cites work: Directed Abelian Groups, Countably Continuous Rings, and Rickart C* -Algebras
cites work: Lifting units in self-injective rings and an index theory for Rickart \(C^ *\)-algebras
description: scientific article; zbMATH DE number 4002304