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Author: Eben Matlis
Publisher: University of Chicago Press
ISBN: 9780226510743
Category : Mathematics
Languages : en
Pages : 180
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Book Description
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.
Author: Eben Matlis
Publisher: University of Chicago Press
ISBN: 9780226510743
Category : Mathematics
Languages : en
Pages : 180
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Book Description
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.
Author: Eben Matlis
Publisher: American Mathematical Soc.
ISBN: 0821812491
Category : Modules
Languages : en
Pages : 70
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Book Description
Author: Joseph J. Rotman
Publisher: Springer Science & Business Media
ISBN: 0387683240
Category : Mathematics
Languages : en
Pages : 722
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Book Description
Graduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology. In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author has also included material about homotopical algebra, alias K-theory. Learning homological algebra is a two-stage affair. First, one must learn the language of Ext and Tor. Second, one must be able to compute these things with spectral sequences. Here is a work that combines the two.
Author: Theodore G. Faticoni
Publisher: CRC Press
ISBN: 9781584887270
Category : Mathematics
Languages : en
Pages : 344
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Book Description
With plenty of new material not found in other books, Direct Sum Decompositions of Torsion-Free Finite Rank Groups explores advanced topics in direct sum decompositions of abelian groups and their consequences. The book illustrates a new way of studying these groups while still honoring the rich history of unique direct sum decompositions of groups
Author: Marco Fontana
Publisher: Springer
ISBN: 3319658743
Category : Mathematics
Languages : en
Pages : 374
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Book Description
This volume presents a collection of articles highlighting recent developments in commutative algebra and related non-commutative generalizations. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non-Noetherian ring theory, module theory and integer-valued polynomials along with connections to algebraic number theory, algebraic geometry, topology and homological algebra. Most of the eighteen contributions are authored by attendees of the two conferences in commutative algebra that were held in the summer of 2016: “Recent Advances in Commutative Ring and Module Theory,” Bressanone, Italy; “Conference on Rings and Polynomials” Graz, Austria. There is also a small collection of invited articles authored by experts in the area who could not attend either of the conferences. Following the model of the talks given at these conferences, the volume contains a number of comprehensive survey papers along with related research articles featuring recent results that have not yet been published elsewhere.
Author: Piotr A. Krylov
Publisher: Walter de Gruyter
ISBN: 9783110200539
Category : Mathematics
Languages : en
Pages : 376
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Book Description
"In this book, modules over a specific class of rings, the discrete valuations domains, are considered. Such modules call for a special consideration, since they have specific properties and play an important role in various areas of algebra, especially of commutative algebra. The text is accompanied by exercises, historical remarks, links to related fields and open problems. It is useful for students, graduates studying algebra, young researchers, and experts."--BOOK JACKET.
Author: P.C. Eklof
Publisher: Elsevier
ISBN: 0080527051
Category : Mathematics
Languages : en
Pages : 620
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Book Description
This book provides a comprehensive exposition of the use of set-theoretic methods in abelian group theory, module theory, and homological algebra, including applications to Whitehead's Problem, the structure of Ext and the existence of almost-free modules over non-perfect rings. This second edition is completely revised and udated to include major developments in the decade since the first edition. Among these are applications to cotorsion theories and covers, including a proof of the Flat Cover Conjecture, as well as the use of Shelah's pcf theory to constuct almost free groups. As with the first edition, the book is largely self-contained, and designed to be accessible to both graduate students and researchers in both algebra and logic. They will find there an introduction to powerful techniques which they may find useful in their own work.
Author: Adnan Tercan
Publisher: Birkhäuser
ISBN: 3034809522
Category : Mathematics
Languages : en
Pages : 389
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Book Description
The main focus of this monograph is to offer a comprehensive presentation of known and new results on various generalizations of CS-modules and CS-rings. Extending (or CS) modules are generalizations of injective (and also semisimple or uniform) modules. While the theory of CS-modules is well documented in monographs and textbooks, results on generalized forms of the CS property as well as dual notions are far less present in the literature. With their work the authors provide a solid background to module theory, accessible to anyone familiar with basic abstract algebra. The focus of the book is on direct sums of CS-modules and classes of modules related to CS-modules, such as relative (injective) ejective modules, (quasi) continuous modules, and lifting modules. In particular, matrix CS-rings are studied and clear proofs of fundamental decomposition results on CS-modules over commutative domains are given, thus complementing existing monographs in this area. Open problems round out the work and establish the basis for further developments in the field. The main text is complemented by a wealth of examples and exercises.
Author: Mary Turgi
Publisher:
ISBN:
Category : Modules (Algebra)
Languages : en
Pages : 332
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Book Description
Author: Paul E. Bland
Publisher: Wiley-VCH
ISBN:
Category : Mathematics
Languages : en
Pages : 164
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Book Description
Torsion theory provides an umbrella under which many classical properties of rings and modules can be reformulated. The purpose of this book is to provide the reader with a quick introduction to torsion theory and to study selected properties of rings and modules in this setting. The material presented ranges from a torsion theoretical treatment of standard topics in ring and module theory to how previously untreated properties of rings and modules might be dealt with in this setting. The approach has been to develop the material so that classical results can be recovered by selecting an appropriate torsion theory. Simple modules, maximal submodules, the Jacobson radical and modules with chain conditions are investigated relative to a torsion theory. A relative form of Nakayama’s lemma is given and a generalized Hopkins-Levitzki theorem is established. Injective and projective concepts are studied and a generalized Baer’s condition for injective modules and a generalized Fuchs condition for quasi-injective modules are shown to hold. Flat modules and covers and hulls of modules are investigated and the concept of a relative (quasi-)projective cover is used to characterize those rings which are right perfect modulo their torsion ideal. Torsion free covers are also studied and results are given which generalize well known results on torsion free covers for modules (with usual torsion) over an integral domain. Finally, (semi-)primitive, (semi-)simple and (semi-)prime rings are investigated in a torsion theoretical setting and rings which are right primitive relative to a torsion theory are linked to a form of density which is reminiscent of Jacobson density for right primitive rings.
