Modules over Discrete Valuation Domains
This book contains a lot of interesting material on modules over discrete valuation domains (DVDs). The first two chapters are introductory and contain an exposition of standard ring and module theory notions (injectivity, purity and completions in the context of modules over a DVD). Chapters 3 and 4 bring some known results on endomorphism rings of modules over a (complete) DVD; in particular the Jacobson radical and the radical factor are calculated for the endomorphism ring of a complete torsion-free module and the Harrison-Matlis equivalence for the endomorphism ring of a divisible primary module is used. The fourth chapter is devoted to the problem of realization of an abstract ring as an endomorphism ring of a module over a DVD; finite topology plays an important role. Such results are used to construct various examples of modules over a DVD with interesting behaviour regarding the direct sum decompositions.
Chapters 5-8 restrict to the case of a commutative DVD. The structure of torsion-free and mixed modules over such rings is studied; some of the techniques resemble those for Abelian groups (passing to the category of quasi-homomorphisms). Among other things, some cardinal invariants providing information for the classification of Warfield modules are defined. Chapter 7 studies when modules over a commutative DVD with isomorphic endomorphism rings are isomorphic. The last chapter is devoted to modules having many endomorphisms or automorphisms. These modules can be seen as a generalization of divisible modules. The book is accessible to any reader, with the only prerequisites being elementary algebra and topology (although some notions like extension groups and inverse limits are referred to other texts). Chapters are concluded by historical remarks, suggestions for further reading, exercises of varying difficulty and open problems.