Rank factorization and bordering of regular matrices over commutative rings
type: Item
instance of: scholarly article
title: Rank factorization and bordering of regular matrices over commutative rings
zbMATH Open document ID: 0951.15012
DOI: 10.1016/S0024-3795(99)00236-0
author: E. Ballico
published in: Linear Algebra and its Applications
publication date: 2000-12-06T00:00:00Z
review text: Let \(R\) be a commutative ring and \(A\) be an \(n \times m\) matrix over \(R\). An elementary extension of \(A\) is a matrix denoted by \(A(\alpha,\beta)\) for some (non-negative) integers \(\alpha\) and \(\beta\), which is of size \((m+\alpha+\beta)\times (n+\beta)\) with \(A\) in the left corner, the \(\beta \times \beta\) identity matrix along the diagonal and with 0 in the remaining places. The author shows that if the maximal spectrum of \(R\) is a Noetherian space of dimension \(d\) and \(A\) is a regular \(m \times n\) matrix of rank \(r\) whose associated module \(Ker(A)\) is stably free, then \(A(\alpha,\beta)\) has a rank factorization with \(\alpha= \max(d+1+r-n,0)\) and \(\beta= \max(d+1-r,0)\). The author also discusses the existence of \(f\) in \(R\) such that borderings and rank factorizations of regular matrices over \(R\) are true allowing \(f\) as denominator. The paper under review has its origin in the work of \textit{K. Manjunatha Prasad} and \textit{K. P. S. Bhaskara Rao} [Linear Algebra Appl. 234, 245-259 (1996; Zbl 0841.15009)].
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 1419722
zbMATH Keywords: rank factorization
zbMATH Keywords: bordering
zbMATH Keywords: regular matrix
zbMATH Keywords: commutative ring
zbMATH Keywords: elementary extension of a matrix
zbMATH Keywords: stably free module
zbMATH Keywords: universal denominator
zbMATH Keywords: maximal spectrum
zbMATH Keywords: Noetherian space
MaRDI profile type: Publication
cites work: Generalized inverses over integral domains
cites work: Q4143433
cites work: Serre's conjecture
cites work: On bordering of regular matrices
full work available at URL: s0024-3795(99)00236-0
OpenAlex ID: W2091739475
Wikidata QID: Q126399801
description: scientific article; zbMATH DE number 1419722