No abstract

Let A and B be rings and Ma and Ng a right A-module and a right 5-module respectively. Let a:End(M,4) -»• End(Ar fl ) be either an isomorphism or an anti-isomorphism of their endomorphism rings. The Wedderburn-Baer-Kaplansky problem is to characterize all such a. The problem has a long history. The aim of these lectures is to consider main results in this direction for modules Ma and Ng which are close to free ones (in the epicenter we put strong generators, i.e., modules with a cyclic free direct summand). The problems under consideration are connected with conditions for inducing of isomorphisms of endomorphism rings by semilinear transformations, by Morita equivalences or by full embeddings. We consider also multiplicative isomorphisms of endomorphism semigroups of modules. We touch some related topics. First of all, we describe recent results on isomorphisms of general linear groups over arbitrary associative rings (Schreier -van der Waerden problem), because of a nice reduction of this group problem to the ring problem of classification of isomorphisms and anti-isomorphisms of matrix rings. We consider Malcev type theorems on elementary equivalences of linear groups and rings of matrices. At the same time we deal with the characterization problem of endomorphism rings of modules from different classes (in particular, of matrix rings). 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 .