Influence of the Baer–Kaplansky theorem on the development of the theory of groups, rings, and modules

The review presents an analysis of results of the Abelian group theory, as well as rings and modules, which concern the definability of algebraic structures by their endomorphism rings and related structures. In the systematization of the results, the greatest attention is paid to torsion-free Abelian groups, which are of particular interest due to the presence of non-isomorphic direct decompositions in this class. This significantly expands the understanding of general, including modern, trends of the development of algebra in the context related to the Baer–Kaplansky theorem.
The reflection of the properties of algebraic objects of a certain class in their endomorphism rings is a natural structural connection, the study of which is a separate investigation direction. A striking introduction to this topic was the Baer–Kaplansky theorem for torsion Abelian groups, which dates back to the middle of the last century and states that any isomorphism of endomorphism rings of two groups from this class is inevitably induced by some isomorphism of the groups themselves. Of course, it follows that if two torsion Abelian groups have isomorphic endomorphism rings, then they are isomorphic. This profound result inspired mathematicians to obtain results in the same form concerning other classes of objects. But even in the theory of Abelian groups itself, other classes were discovered for which the analogue of the Baer–Kaplansky theorem is valid. Despite the fundamental difference between the definitions of completely decomposable Abelian groups, which are direct sums of rank-one torsion-free groups, and torsion Abelian groups considered, which are direct sums of finite order cyclic groups, there is one very important common characteristic of these classes: their decompositions into indecomposable summands are determined uniquely up to isomorphism. This property is not possessed by torsion-free Abelian groups in general, whose definability by their endomorphism rings is in the focus of our attention.