Generalizing the Baer-Kaplansky theorem
The well-known Baer-Kaplansky theorem states that two Abelian torsion groups are isomorphic if (and only if) their endomorphism rings are isomorphic. The author generalizes this theorem to modules over non-commutative rings, by introducing some new concepts and suitably modifying the formulation of the theorem. For two modules \(M\) and \(N\), a ring isomorphism \(f\) from \(\text{End}(M)\) to \(\text{End}(N)\) is called an IP-isomorphism if for every primitive idempotent \(e\in\text{End}(M)\), one gets that \(f(e)N\) is isomorphic to \(eM\). It is shown in the paper that, under certain additional hypotheses, two modules \(M\) and \(N\) (over any ring \(R\)) that are direct sums of indecomposable modules are isomorphic if and only if there exists an IP-isomorphism between their endomorphism rings. The author also relates the Baer-Kaplansky theorem to a statement about rings satisfying the property that every automorphism is an inner IP-isomorphism. The problem of characterizing rings satisfying this property is posed in the paper, and some preliminary results are obtained.
scientific article; zbMATH DE number 1308268