IBN and related properties for rings
type: Item
instance of: scholarly article
title: IBN and related properties for rings
zbMATH Open document ID: 1012.16007
DOI: 10.1023/A:1015683326841
author: K. Varadarajan
author: Ahmad Haghany
published in: Acta Mathematica Hungarica
publication date: 2003-05-18T00:00:00Z
review text: Let \(R\) be an associative ring with \(1\neq 0\). Then \(R\) has invariant basis number (IBN) if \(R^m\cong R^n\) in \(R\)-mod implies that \(m=n\), and \(R\) has the rank condition if \(R^m\to R^n\to 0\) exact in \(R\)-mod implies that \(m\geq n\). If \(R=R_1\times R_2\) then (1) \(R\) fails to have IBN if and only if each \(R_i\) fails to have IBN, and (2) \(R\) fails to have the rank condition if and only if each \(R_i\) fails to have the rank condition. A formal \(2\times 2\) block triangular matrix ring \(T\) has IBN (rank condition) if and only if one of the diagonal blocks has IBN (rank condition); \(T\) is stably finite if and only if both of its diagonal blocks are stably finite. Stable finiteness is a Morita invariant property.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16U20
zbMATH DE Number: 1912016
zbMATH Keywords: IBN-rings
zbMATH Keywords: stably finite rings
zbMATH Keywords: (strong) rank condition
zbMATH Keywords: Morita invariant
zbMATH Keywords: triangular matrix rings
reviewed by: Mark L. Teply
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1912016