\(K\)-theoretically simple von Neumann regular rings
type: Item
instance of: scholarly article
title: \(K\)-theoretically simple von Neumann regular rings
zbMATH Open document ID: 0837.16008
author: Pere Ara
author: Kenneth R. Goodearl
author: D. V. Tyukavkin
author: Enrique Pardo Espino
published in: Journal of Algebra
publication date: 1996-04-24T00:00:00Z
review text: The authors investigate the differences between simplicity of a von Neumann regular ring and simplicity of its ordered Grothendieck group \(K_0\). After giving preliminaries in \S 1, they derive properties of pseudo-rank functions on ideals in a regular ring. In \S 3 they prove that if \(R\) is a stably finite, \(K_0\)-simple, non-Artinian regular ring and if the center of \(R\) is uncountable, then \(R\) is a simple unit-regular ring. They also prove that if \(R\) is a stably finite regular ring without nonzero Artinian homomorphic images then \(R\) is simple if and only if \(R\) satisfies approximate weak \(N^*\)-comparability. The authors show that given a countable field \(F\) and a countable dimension group \((G,u)\) with order-unit, there exists a stably finite, \(K_0\)-simple, regular \(F\)- algebra \(R\) such that (i) \(R\) has unique minimal nonzero ideal \(J\) and the lattice of ideals of \(R/J\) is isomorphic to the lattice of ideals of \(G\), (ii) The center of \(R\) is \(F\) and (iii) \(R\) has a unique rank function. Finally in \S 4, they prove that if \(R\) is a simple regular ring, then \(R\) satisfies weak comparability if and only if the category \(FP(R)\) of all finitely generated projective \(R\)-modules is strictly unperforated.
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 783882
zbMATH Keywords: simplicity
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: ordered Grothendieck groups
zbMATH Keywords: pseudo-rank functions
zbMATH Keywords: simple unit-regular rings
zbMATH Keywords: stably finite regular rings
zbMATH Keywords: approximate weak \(N^*\)-comparability
zbMATH Keywords: lattice of ideals
zbMATH Keywords: finitely generated projective \(R\)-modules
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: jabr.1995.1145
OpenAlex ID: W2012526261
DOI: 10.1006/JABR.1995.1145
description: scientific article; zbMATH DE number 783882