Around the Baer-Kaplansky theorem
type: Item
instance of: scholarly article
title: Around the Baer-Kaplansky theorem
zbMATH Open document ID: 1487.16002
author: A. V. Tsarev
author: P. A. Krylov
author: A. A. Tuganbaev
published in: Journal of Mathematical Sciences (New York)
publication date: 2021-06-29T00:00:00Z
review text: Let \(\mathcal M\) be a class of modules. A Baer-Kaplansky theorem for \(\mathcal M\) is the statement that for all \(M\) and \(N\in\mathcal M,\ \)End\((M)\cong \)\ End\((N)\) implies that \(M\cong N\). A strong Baer-Kaplansky theorem is the statement that every isomorphism from End\((M)\) to End\((N)\) is induced by an isomorphism from \(M\) to \(N\). In this largely expository paper, the authors review progress in these theorems over the past 80 years, often with new unified proofs. The original version of the Baer-Kaplansky theorems was in the context of vector spaces over division rings, but the deepest results concern torsion abelian groups and related modules. Related theorems involve replacing the endomorphism rings by their Jacobson radicals, or by replacing an algebraic isomorphism between End\((M)\) and End\((N)\) by an isomorphism bi-continuous with respect to the finite topology. In the latter case, suitable classes \(\mathcal M\) include homogeneous fully transitive torsion-free abelian groups of idempotent type.
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 7364546
zbMATH Keywords: abelian group
zbMATH Keywords: endomorphism ring
zbMATH Keywords: isomorphism theorem for endomorphism rings
zbMATH Keywords: Baer-Kaplansky theorem
zbMATH Keywords: Kaplansky method
reviewed by: Phillip Schultz
MaRDI profile type: Publication
full work available at URL: s10958-021-05428-w
OpenAlex ID: W3173568634
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DOI: 10.1007/S10958-021-05428-W
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description: scientific article; zbMATH DE number 7364546