Direct Sum Decompositions of Torsion-Free Finite Rank Groups
This monograph is devoted to direct sum decompositions of reduced torsion free finite rank (rtffr) Abelian groups. Any Abelian group G of finite rank has an indecomposable decomposition, hence one can study questions related to the uniqueness of indecomposable decompositions of G. The book offers a technique that passes these problems to study the factor of End(G) modulo its nilradical N(End(G)). Let us denote this ring by E(G). The book provides a lot of interesting results not included in other books. The first chapter contains some preliminaries. The second chapter explains some motivation. The Krull-Schmidt-Azumaya and the Baer-Kulikov-Kaplansky theorems are explained as examples of good behaviour. On the other hand, the Corner result and other constructions are mentioned to show almost arbitrarily bad behaviour of direct-sum decompositions of rtffr Abelian groups. The notion of quasi-isomorphism and local isomorphism is introduced and the Jόnsson theorem shows that one gets much better behaviour when considering this problem up to a quasi-isomorphism. Chapter 3 explains how the local isomorphism classes of finitely generated projective modules over a semiprime ring having its additive group rtffr are translated into isomorphism classes of finitely generated projectives over a different ring. The next chapter gives results when some commutativity conditions in End(G) are satisfied.
The fifth chapter investigates what can be said about the number of isomorphism classes of groups locally isomorphic to a strongly indecomposable rtffr group with E(G) being a commutative domain. Chapter 6 studies the Baer splitting property. In chapter 8, the author studies Gabriel filters, in particular the filter of divisibility and its relation to the quasi-splitting of some exact sequences. The last chapter returns to E-properties and possible values of homological dimensions of G over End(G) are discussed. The reader of this book is supposed to be rather advanced in Abelian groups. The exposition is almost self-contained, with just a few results going far beyond the scope of the book (for example results from analytic number theory) and which are stated without proofs. A lot of examples illustrate the theory. The author provides a number of exercises and suggestions for further research at the end of each chapter.