## Modules over Non-Noetherian DomainsIn this book, the authors present both traditional and modern discoveries in the subject area, concentrating on advanced aspects of the topic. Existing material is studied in detail, including finitely generated modules, projective and injective modules, and the theory of torsion and torsion-free modules. Some topics are treated from a new point of view. Also included are areas not found in current texts, for example, pure-injectivity, divisible modules, uniserial modules, etc. Special emphasis is given to results that are valid over arbitrary domains. The authors concentrate on modules over valuation and Prufer domains, but also discuss Krull and Matlis domains, $h$-local, reflexive, and coherent domains. The volume can serve as a standard reference book for specialists working in the area and also is a suitable text for advanced-graduate algebra courses and seminars. |

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### Contents

1 | |

Chapter II Valuation Domains | 57 |

Chapter III Prüfer Domains | 91 |

Chapter IV More NonNoetherian Domains | 123 |

Chapter V Finitely Generated Modules | 149 |

Chapter VI Projectivity and Projective Dimension | 195 |

Chapter VII Divisible Modules | 247 |

Chapter VIII Topology and Filtration | 273 |

Chapter XII Polyserial Modules | 403 |

Chapter XIII RDand PureInjectivity | 423 |

Chapter XIV Torsion Modules | 463 |

Chapter XV TorsionFree Modules of Finite Rank | 489 |

Chapter XVI Infinite Rank TorsionFree Modules | 531 |

Appendix on Set Theory | 585 |

591 | |

603 | |

Chapter IX Injective Modules | 305 |

Chapter X Uniserial Modules | 337 |

Chapter XI Heights Invariants and Basic Submodules | 381 |

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### Common terms and phrases

abelian group algebraic annihilated uniserial assume Bézout domain cardinal commutative consequence contains continuous well-ordered ascending cotorsion countably Dedekind domains defined denote direct sum divisible modules divisorial element endomorphism ring equivalent exact sequence Exercises exists Exth finite rank finitely generated modules finitely presented fractional ideal free R-module h-divisible h-local Hence henselian Hint Homş homomorphism hypothesis idempotent implies indecomposable induced injective module integrally closed isomorphic isomorphy classes Lemma linearly compact marimal maximal ideal noetherian non-finitely annihilated non-standard uniserial non-zero polynomial polyserial module presented modules prime ideal projective dimension Proof Proposition prove Prüfer domain pure submodule pure-injective quotients R-complete R-module RD-injective RD-submodule result satisfies standard uniserial subset sum of cyclic summand Theorem topology torsion module torsion-free modules totally ordered uniserial modules valuation domain valuation ring well-ordered ascending chain