The subject of torsion-free modules over an arbitrary integral domain arises naturally as a generalization of torsion-free abelian groups. In this volume, Eben Matlis brings together his research on torsion-free modules that has appeared in a number of mathematical journals. Professor Matlis has reworked many of the proofs so that only an elementary knowledge of homological algebra and commutative ring theory is necessary for an understanding of the theory.
The first eight chapters of the book are a general introduction to the theory of torsion-free modules. This part of the book is suitable for a self-contained basic course on the subject. More specialized problems of finding all integrally closed D-rings are examined in the last seven chapters, where material covered in the first eight chapters is applied.
An integral domain is said to be a D-ring if every torsion-free module of finite rank decomposes into a direct sum of modules of rank 1. After much investigation, Professor Matlis found that an integrally closed domain is a D-ring if, and only if, it is the intersection of at most two maximal valuation rings.
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Noetherian reflexive rings
Completely reflexive rings
Rings of type I
Integrally closed Drings
Applying the functor assume completely reflexive ring contained contradiction shows cotorsion R-module D-ring direct sum direct summand discrete valuation ring domain of Krull element r e epimorphism exact sequence exists an element Ext Q finite rank Gorenstein ring h-local ring h-reduced hence by Theorem Hence there exists ideal of F indecomposable injective envelope integral closure integral domain isomorphic Krull dimension lemma for Theorem Let F maximal ideal maximal valuation ring module monomorphism Noetherian ring non-zero element nonzero ideal nonzero prime ideal principal ideal Proof Prufer ring pure submodule quasi-local ring R-module of finite R-module of rank R-submodule of Q R-topology R/P-module remote quotient field ring by Theorem ring of Krull ring of type ring with maximal statements are equivalent sufficient to prove Suppose Theorem 29 Theorem 42 Theorem 65 Theorem 83 topology torsion R-module torsion-free R-module torsionless ring