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Isomorphisms and anti-isomorphisms of endomorphism rings of modules. (English) Zbl 0881.16017

Fong, Y. (ed.) et al., First international Tainan-Moscow algebra workshop. Proceedings of the international conference, Tainan, Taiwan, Republic of China, July 23–August 22, 1994. Berlin: de Gruyter. 69-122 (1996).
Let \(M_A\) and \(N_B\) be a right \(A\)-module and a right \(B\)-module for some rings \(A\) and \(B\), respectively. Let \(\alpha\colon\text{End}(M_A)\to\text{End}(N_B)\) be either an isomorphism or an anti-isomorphism of their endomorphism rings. The Wedderburn-Baer-Kaplansky problem is to characterize all such \(\alpha\). The aim of this paper is to consider main results in this direction for modules \(M_A\) and \(N_B\) that are close to free ones. The problems under consideration are connected with conditions for inducing isomorphisms of endomorphism rings by semilinear transformations, by Morita equivalences or by full embeddings. Multiplicative isomorphisms of endomorphism semigroups of modules are considered, too. The author describes recent results on isomorphisms of general linear groups over arbitrary associative rings and considers Mal’cev type theorems on elementary equivalences of linear groups and rings of matrices. Some comments are related to operator rings and semigroups of topological and ordered linear spaces and to Frobenius type results on \(A\)-linear mappings on matrix rings \(A_n\).
The paper contains 1. Introduction; 1.1. Isomorphisms of endomorphism rings; 1.2. Anti-isomorphisms of endomorphism rings; 1.3. Some comments on endomorphism rings and semigroups; 1.4. Characterization of endomorphism rings; 1.5. Books and surveys; 1.6. Acknowledgments; 2. Strong generators; 3. Isomorphisms of endomorphism rings of strong generators; 4. Anti-isomorphisms of endomorphism rings of strong generators; 5. Endomorphism semigroups of modules; 6. Isomorphisms of the general linear group \(GL_N(A)\) over an associative ring \(A\); 7. On Mal’cev type theorems on elementary equivalences of linear groups and rings of matrices; 8. Related topics; 8.1. Representation of rings as full matrix rings (hidden matrices); 8.2. Isomorphisms and automorphisms of triangular and structural matrix rings; 8.3. Isomorphisms and automorphisms of incidence rings and algebras; 8.4. Isomorphisms and anti-isomorphisms of operator rings and semigroups of Banach and topological linear spaces; 8.5. Automorphisms and anti-automorphisms of endomorphism semigroups of partially ordered linear spaces; 8.6. Frobenius type results. There are 298 references.
For the entire collection see [Zbl 0856.00015].

MSC:

16S50 Endomorphism rings; matrix rings
16W20 Automorphisms and endomorphisms
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
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