Automorphismengruppen freier Moduln von unendlichem Rang. (Automorphism groups of free modules of infinite rank)
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.
scientific article; zbMATH DE number 4109018
scientific article