Automorphismengruppen freier Moduln von unendlichem Rang. (Automorphism groups of free modules of infinite rank)
type: Item
instance of: scholarly article
title: Automorphismengruppen freier Moduln von unendlichem Rang. (Automorphism groups of free modules of infinite rank)
zbMATH Open document ID: 0677.20035
DOI: 10.1016/0021-8693(89)90236-6
author: Bertold Laschinger
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let R be an associative ring with 1, \(\Omega\) an infinite set, M the free left R-module spanned by \(\Omega\), \(M^*\) the dual module, E(\(\Omega\),R) the subgroup of GL(\(\Omega\),R) generated by all automorphisms \(m\to m+(m\mu)e\) with \(\mu \in M^*\), \(e\in \Omega\). A subgroup H of E(\(\Omega\),R) is called an SL-group if for any finite subset \(\psi\) of \(\Omega\) the restriction to \(\psi\) of the stabilizer of \(\psi\) in H contains E(\(\psi\),R). It is proved that those H are exactly the stabilizers in E(\(\Omega\),R) of the submodules \(\alpha^*(H)\) of \(M^*\) such that \(\alpha^*(H)\) separates the elements of \(\Omega\). In the second part of this paper, all normal subgroups of SL-groups are classified. This generalizes results of Bass, Arrell, Robertson, Rosenberg. Finally, under restrictions on R, infinitely many non- isomorphic SL-groups are exhibited.
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 4109018
zbMATH Keywords: free left R-module
zbMATH Keywords: automorphisms
zbMATH Keywords: stabilizers
zbMATH Keywords: normal subgroups of SL- groups
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90236-6
OpenAlex ID: W2082279803
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description: scientific article; zbMATH DE number 4109018