Properties of endomorphism rings of Abelian groups. II.
This paper contains the final three chapters of what is now a book: Endomorphism Rings of Abelian Groups [Algebras and Applications 2. Boston MA: Kluwer Academic Publishers (2003; Zbl 1044.20037)]. The first four chapters were published as a separate paper [part I, J. Math. Sci., New York 112, No. 6, 4598-4735 (2002; Zbl 1041.20039)]. Given an Abelian group \(A\), we can form the ring of endomorphisms \(\text{End}_\mathbb{Z}(A)\) and then view the group \(A\) as an \(\text{End}_\mathbb{Z}(A)\)-module in a natural way. A wide variety of tools are available for studying such objects, and the reader of ``Endomorphism Rings'' is assumed to have some familiarity with the theories of Abelian groups, rings, and modules. Anyone with the prerequisites will enjoy the variety of methods, the beauty of the results, and the appeal of the open problems compiled by the authors. As background, I'll list the first four chapters of the book: Chapter I (General Results on Endomorphism Rings). Chapter II (Groups as Modules over Their Endomorphism Rings). Chapter III (Ring Properties of Endomorphism Rings). Chapter IV (Jacobson Radical). Chapter V (Isomorphism or Realization Theorems for Endomorphism Rings) is a discussion of the relationship between two groups whose endomorphism rings are isomorphic, and of which rings can be ``realized'' as endomorphism rings of various kinds of groups. The Baer-Kaplansky Theorem for torsion Abelian groups and Corner's Theorem on countable, reduced torsion-free rings are perhaps the most well-known results in these areas. Due to the limitations of space, the chapter omits the extensive literature on realizing rings as endomorphism rings of ``large'' Abelian groups using the Black Box of Shelah and related combinatorial arguments. Chapter VI (Hereditary Endomorphism Rings and Related Topics). A ring \(R\) is called right hereditary if each right ideal is projective as a right \(R\)-module. The importance of this property for endomorphism rings of Abelian groups has been demonstrated repeatedly. In particular, some classical theorems of Baer on homogeneous completely decomposable groups remain true if rank-1 groups are replaced by groups with hereditary endomorphism rings. Other important notions studied in this chapter include `self small', `faithful' and `endoflat'. The \(\Hom\)-\(\otimes\) category equivalence between categories of \(A\)-projective groups and categories of projective modules over \(\text{End}_\mathbb{Z}(A)\) is presented in its various forms. Chapter VII (Completely Transitive Groups and Endomorphism Rings). A torsion-free group \(G\) is called `completely transitive' if for any two nonzero elements \(a,b\in G\) with \(\text{height}(a)\leq\text{height}(b)\), there is an endomorphism of \(G\) mapping \(a\) to \(b\). The group is called `transitive' if `endomorphism' is replaced by `automorphism' in the definition. The class of completely transitive groups contains all algebraically compact groups and all homogeneous separable groups, along with the quasi-pure injective and strongly homogeneous groups. The chapter also discusses other important subclasses of completely transitive groups, such as those with maximum condition on types. -- The paper concludes with a list of 451 references. The material is largely self-contained, although some proofs are given by reference when space becomes an issue. Helpful and challenging sets of exercises appear at the end of each section and open problems are listed at the end of each chapter. This work is valuable as a text for a graduate course, as a reference on a lively area of mathematics, and as a guide to appealing research problems.
scientific article; zbMATH DE number 1916225