Around the Baer-Kaplansky theorem
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.
isomorphism theorem for endomorphism rings
scientific article; zbMATH DE number 7364546
scientific article