## Modules Over Discrete Valuation DomainsThe aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Jacobs University, Bremen, GermanyKatrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021) |

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### Contents

4 | 27 |

Endomorphism rings of divisible and complete modules | 86 |

free modules | 126 |

Mixed modules | 203 |

Determinity of modules by their endomorphism rings | 272 |

Modules with many endomorphisms or automorphisms | 308 |

Symbols | 351 |

### Other editions - View all

Modules Over Discrete Valuation Domains Piotr A. Krylov,Askar A. Tuganbaev No preview available - 2008 |

### Common terms and phrases

Abelian addition algebra arbitrary assertion assume automorphism basic submodule basis called Chapter clear coincides complete complete module Consequently consider contained Corollary corresponding cotorsion cyclic modules decomposition defined definition denote direct sum direct summand discrete valuation domain divisible divisible module element embedding End(M endomorphism rings equal equivalent exact sequence example Exercise exists extended factor field finite finite rank follows free module fully transitive height holds Hom(M homomorphism ideal indecomposable module independent induced infinite invariants invertible element isomorphism lattice Lemma mapping maximal mixed modules morphism multiplication nonzero object obtain ordinal number positive integer primary module problem Proof Proposition prove pure submodule R-module rank relation respect satisfies Section sequence similar space Theorem theory topology torsion torsion-free module unique verify