Journal of Mathematical Sciences, Vol. 145, No. 4, 2007 MODULES OVER DISCRETE VALUATION DOMAINS. I P. A. Krylov and A. A. Tuganbaev UDC 512.553 CONTENTS Chapter 1. Preliminaries ......................................... 4998 1. Some Definitions and Notation ................................... 4998 2. Endomorphisms and Homomorphisms of Modules ......................... 5002 3. Discrete Valuation Domains ..................................... 5008 4. Primary Notions of Module Theory ................................. 5018 Chapter 2. Basic Facts .......................................... 5027 5. Free Modules ............................................. 5027 6. Divisible Modules ........................................... 5029 7. Pure Submodules ........................................... 5033 8. Direct Sums of Cyclic Modules ................................... 5036 9. Basis Submodules .......................................... 5040 10. Pure-Projective and Pure-Injective Modules ............................ 5044 11. Complete Modules .......................................... 5048 Chapter 3. Endomorphism Rings of Divisible and Complete Modules ................ 5053 12. Examples of Endomorphism Rings ................................. 5054 13. Harrison–Matlis Equivalence .................................... 5055 14. The Jacobson Radical ........................................ 5059 15. Galois Correspondences ....................................... 5066 Chapter 4. Representation of Rings by Endomorphism Rings .................... 5073 16. The Finite Topology ......................................... 5074 17. The Ideal of Finite Endomorphisms ................................ 5075 18. Characterization Theorems for Endomorphism Rings of Torsion-Free Modules ........ 5080 19. Realization Theorems for Endomorphism Rings of Torsion-Free Modules ............ 5085 20. Essentially Indecomposable Modules ................................ 5090 21. Cotorsion Modules and Cotorsion Hulls .............................. 5094 22. An Embedding from the Category of Torsion-Free Modules into the Category of Mixed Modules .......................................... 5100 References .................................................. 5106 Introduction Discrete valuation domains form a class of local domains close to division rings. However, it follows from our definition that a division ring is not a discrete valuation domain. A discrete valuation domain has a unique (up to an invertible factor) prime element. A commutative discrete valuation domain is a principal ideal domain with unique prime element. Therefore, the role of prime elements is very important in the theory of modules over discrete valuation domains. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 121, Algebra, 2006. 1072–3374/07/1454–4997 c  2007 Springer Science+Business Media, Inc. 4997