Work of Pere Menal on normal subgroups
type: Item
instance of: scholarly article
title: Work of Pere Menal on normal subgroups
zbMATH Open document ID: 0790.20064
DOI: 10.5565/PUBLMAT_362A92_03
author: F. A. Arlinghaus
author: L. N. Vaserstein
published in: Publicacions Matemàtiques
publication date: 1994-07-04T00:00:00Z
full work available at URL: https://eudml.org/doc/41719
review text: The authors survey the work of the late Pere Menal on the subgroup structure of \(\mathrm{GL}_ 2A\) for rings \(A\) of various types. For a large class of rings \(A\), which includes all commutative, von Neumann regular and stable range 1 rings, together with all Banach algebras, it is known that, when \(n \geq 3\), the subgroups of \(\mathrm{GL}_ nA\) normalized by \(E_ nA\) (the subgroup generated by the elementary matrices) can be completely classified in terms of the (two-sided) \(A\)-ideals. For anything like such a classification to extend to the case \(n = 2\) further restrictions usually have to be imposed on \(A\). (Any classification of this type appears to fail, for example, when \(A = \mathbb Z\).) Menal (in collaboration with the second author) has classified the \(E_ 2A\)-normalized subgroups of \(\mathrm{GL}_ 2A\) when (i) \(A\) is a von Neumann regular ring, (ii) \(A\) is a Banach algebra or a stable range 1 ring containing \(1/2\), and (iii) \(A\) is a local ring, where \(A/\text{rad }A\) has at least 4 elements. In case (i) the classification involves the \(A\)-ideals. For (ii), (iii) the classification involves the quasi-ideals of \(A\). The principal way in which these results improve on the (many) previously known special cases is that in nearly all of the latter the ring is assumed to be commutative.
reviewed by: A. W. Mason
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20-03
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 203304
zbMATH Keywords: two-dimensional general linear groups
zbMATH Keywords: subgroup structure of \(GL_ 2A\)
zbMATH Keywords: elementary matrices
zbMATH Keywords: \(E_ 2A\)-normalized subgroups
zbMATH Keywords: von Neumann regular ring
zbMATH Keywords: Banach algebra
zbMATH Keywords: stable range 1 ring
zbMATH Keywords: local ring
zbMATH Keywords: quasi-ideals
MaRDI profile type: Publication
OpenAlex ID: W2026537682
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description: scientific article; zbMATH DE number 203304