Products of commutators of transvections over local rings
type: Item
instance of: scholarly article
title: Products of commutators of transvections over local rings
zbMATH Open document ID: 1014.20022
DOI: 10.1016/S0024-3795(02)00364-6
author: Baodong Zheng
author: Hong You
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: Let \(R\) be a commutative local ring, and \(M\) the maximal ideal of \(R\). The natural map \(\pi\colon R\to R/M\) will induce the natural map \(R^n\to(R/M)^n\). For any matrix \(A\) in \(\text{GL}_nR\), write \(\text{res }A=n-\dim(\pi P)\), where \(P=\{v\in R^n\mid Av=v\}\) is the fixed submodule of \(A\). An invertible matrix \(A\) over \(R\) is a transvection if \(\text{res }A=1\) and \(\det A=1\). The authors show that if \(|R/M|>3\), then every matrix in \(\text{SL}_nR\) (\(n\geq 2\)) is a product of at most \([n/2]+2\) commutators of transvections and the length \([\text{res }A/2]+2\) cannot be reduced for \(n=2\).
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1864768
zbMATH Keywords: products of commutators
zbMATH Keywords: transvections
zbMATH Keywords: local rings
MaRDI profile type: Publication
cites work: Q4043214
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description: scientific article; zbMATH DE number 1864768