Endomorphism Rings of Abelian Groups

Abstract

This paper contains a review of results on endomorphism rings of Abelian groups. On one hand, this rapidly developing section of contemporary algebra can be considered as a part of Abelian group theory; on the other hand, it can be considered as a branch of the theory of endomorphism rings of modules. This section is close to both theories, but it has many specific features. There are several important reasons to study endomorphism rings of Abelian groups. First, it provides us with new information on these groups. Second, it stimulates the study of the theory of modules and their endomorphism rings. There are other fields of algebra where the application of endomorphism rings can be useful (additive groups of rings, E-modules and E-rings, and so on). There are already many excellent results in the theory of endomorphism rings of Abelian groups. A large number of methods are used (for example, group, module, categorical, topological, and set-theoretical methods). This survey gives a satisfactory overview of the content and methods of this part of mathematics. One chapter of the book of L. Fuchs [128] is devoted to endomorphism rings of Abelian groups. Also, they are considered in the works of I. Kaplansky [183], A. G. Kurosh [216], D. Arnold [29], and K. Benabdallah [51]. A number of results in this field of algebra are considered in the surveys of A. P. Mishina [244–248], A. V. Mikhalev [242], A. V. Mikhalev and A. P. Mishina [243], V. T. Markov, A. V. Mikhalev, L. A. Skornyakov, and A. A. Tuganbaev [233]. The work of R. Baer [44] has played an important role in the making of the theory of endomorphism rings of Abelian groups and modules. Several fields of ring theory related to endomorphism rings of modules are considered in the works of I. Lambek [218], C. Faith [109, 110], F. Kasch [184], A. A. Tuganbaev [328, 329], and others. However, there does not exist a book which is especially devoted to endomorphism rings; also, there does not exist a systematical presentation of the main results of this theory. This survey slightly fills in this appreciable gap. We note that the book of L. Fuchs does not reflect all the fields of the theory of endomorphism rings. In addition, new sections of this theory appeared after the publication of this book; also, several excellent results were obtained in traditional sections of the theory. In this paper, we consider the main fields of the theory of endomorphism rings of Abelian groups. The most typical results of this theory are included in the review. Some theorems are restated (some statements are presented in more general form, and some theorems are presented in a simplified form). In Secs. 8 and 9, we use the papers of May [235], Gobel [136], and Corner–Gobel [74]. Some parts of these papers are included in the corresponding sections. Our review is designed for specialists in the theory of Abelian groups and the theory of rings and modules. We wish to avoid routine recounting of results and try to give an idea of the possibilities, methods of proofs, and relations between various research fields and individual results. Unsolved problems are presented at the end of every section. Most of them are known; we only systematize them. The authors do not wish to present a complete bibliography; also, there are difficulties related to precedence. Every Abelian group belongs to exactly one of the following three classes of groups: torsion groups, torsion-free groups, and mixed groups. (A group is said to be mixed if it contains a nonzero element of finite order and an element of infinite order.) The properties of endomorphism rings of groups of these three classes are often different. Usually, we emphasize this fact.