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Modules Over Discrete Valuation Rings

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Modules Over Discrete Valuation Rings

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Author : Piotr A. Krylov
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-09-24

Modules Over Discrete Valuation Rings written by Piotr A. Krylov and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-24 with Mathematics categories.

This book provides the first systematic treatment of modules over discrete valuation domains, which play an important role in various areas of algebra, especially in commutative algebra. Many important results representing the state of the art are presented in the text along with interesting open problems. This updated edition presents new approaches on p-adic integers and modules, and on the determinability of a module by its automorphism group. Contents Preliminaries Basic facts Endomorphism rings of divisible and complete modules Representation of rings by endomorphism rings Torsion-free modules Mixed modules Determinity of modules by their endomorphism rings Modules with many endomorphisms or automorphisms

Modules Over Discrete Valuation Domains

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Author : Piotr A. Krylov
language : en
Publisher: Walter de Gruyter
Release Date : 2008-08-27

Modules Over Discrete Valuation Domains written by Piotr A. Krylov and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-08-27 with Mathematics categories.

This book provides the first systematic treatment of modules over discrete valuation domains which plays an important role in various areas of algebra, especially in commutative algebra. Many important results representing the state of the art are presented in the text which is supplemented by exercises and interesting open problems. An important contribution to commutative algebra.

Modules Over Discrete Valuation Domains

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Author : Piotr A. Krylov
language : en
Publisher: Walter de Gruyter
Release Date : 2008

"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.

Modules Over Valuation Rings

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Author : László Fuchs
language : en
Publisher: CRC Press
Release Date : 1985-03-27

Modules Over Valuation Rings written by László Fuchs and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985-03-27 with Mathematics categories.

This book initiates a systematic, in-depth study of Modules Over Valuation Domains. It introduces the theory of modules over commutative domains without finiteness conditions and examines frontiers of current research in modules over valuation domains. It represents a unique effort to combine ideas from abelian group theory, in a large scale, with powerful techniques developed in module theory. This volume surveys the background material on valuation rings, modules and homological algebra ... features new results for important classes of modules such as finitely generated, divisible, pure-injective, and projective dimension one -- never published before ... contains exercises and research problems -- offering guidance for independent and creative study ... and provides historical notes, comments, and an extensive bibliography. Mathematicians and advanced graduate-level mathematics students interested in module theory, abelian group theory, and commutative ring theory can stay abreast of the latest advances with Modules Over Valuation Domains. Book jacket.

Modules Over Discrete Valuation Rings

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Author : Piotr A. Krylov
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-09-24

This book provides the first systematic treatment of modules over discrete valuation domains, which play an important role in various areas of algebra, especially in commutative algebra. Many important results representing the state of the art are presented in the text along with interesting open problems. This updated edition presents new approaches on p-adic integers and modules, and on the determinability of a module by its automorphism group. ContentsPreliminariesBasic factsEndomorphism rings of divisible and complete modulesRepresentation of rings by endomorphism ringsTorsion-free modulesMixed modulesDeterminity of modules by their endomorphism ringsModules with many endomorphisms or automorphisms

Torsion Free Modules

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Author : Eben Matlis
language : en
Publisher: University of Chicago Press
Release Date : 1972

Torsion Free Modules written by Eben Matlis and has been published by University of Chicago Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972 with Mathematics categories.

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.

Lecture Notes On Local Rings

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Author : Birger Iversen
language : en
Publisher: World Scientific Publishing Company Incorporated
Release Date : 2014

Lecture Notes On Local Rings written by Birger Iversen and has been published by World Scientific Publishing Company Incorporated this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with Mathematics categories.

1. Dimension of a local ring. 1.1. Nakayama's lemma. 1.2. Prime ideals. 1.3. Noetherian modules. 1.4. Modules of finite length. 1.5. Hilbert's basis theorem. 1.6. Graded rings. 1.7. Filtered rings. 1.8. Local rings. 1.9. Regular local rings -- 2. Modules over a local ring. 2.1. Support of a module. 2.2. Associated prime ideals. 2.3. Dimension of a module. 2.4. Depth of a module. 2.5. Cohen-Macaulay modules. 2.6. Modules of finite projective dimension. 2.7. The Koszul complex. 2.8. Regular local rings. 2.9. Projective dimension and depth -- 2.10. [symbol]-depth. 2.11. The acyclicity theorem. 2.12. An example -- 3. Divisor theory. 3.1. Discrete valuation rings. 3.2. Normal domains. 3.3. Divisors. 3.4. Unique factorization. 3.5. Torsion modules. 3.6. The first Chern class. 3.7. Regular local rings. 3.8. Picard groups. 3.9. Dedekind domains -- 4. Completion. 4.1. Exactness of the completion functor. 4.2. Separation of the [symbol]-adic topology. 4.3. Complete filtered rings. 4.4. Completion of local rings. 4.5. Structure of complete local rings -- 5. Injective modules. 5.1. Injective modules. 5.2. Injective envelopes. 5.3. Decomposition of injective modules. 5.4. Matlis duality. 5.5. Minimal injective resolutions. 5.6. Modules of finite injective dimension. 5.7. Gorenstein rings -- 6. Local cohomology. 6.1. Basic properties. 6.2. Local cohomology and dimension. 6.3. Local cohomology and depth. 6.4. Support in the maximal ideal. 6.5. Local duality for Gorenstein rings -- 7. Dualizing complexes. 7.1. Complexes of injective modules. 7.2. Complexes with finitely generated cohomology. 7.3. The evaluation map. 7.4. Existence of dualizing complexes. 7.5. The codimension function. 7.6. Complexes of flat modules. 7.7. Generalized evaluation maps. 7.8. Uniqueness of dualizing complexes -- 8. Local duality. 8.1. Poincare series. 8.2. Grothendieck's local duality theorem. 8.3. Duality for Cohen-Macaulay modules. 8.4. Dualizing modules. 8.5. Locally factorial domains. 8.6. Conductors. 8.7. Formal fibers -- 9. Amplitude and dimension. 9.1. Depth of a complex. 9.2. The dual of a module. 9.3. The amplitude formula. 9.4. Dimension of a complex. 9.5. The tensor product formula. 9.6. Depth inequalities. 9.7. Condition Sr of Serre. 9.8. Factorial rings and condition Sr. 9.9. Condition S[symbol]. 9.10. Specialization of Poincare series -- 10. Intersection multiplicities. 10.1. Introduction to Serre's conjectures. 10.2. Filtration of the Koszul complex. 10.3. Euler characteristic of the Koszul complex. 10.4. A projection formula. 10.5. Power series over a field. 10.6. Power series over a discrete valuation ring. 10.7. Application of Cohen's structure theorem. 10.8. The amplitude inequality. 10.9. Translation invariant operators. 10.10. Todd operators. 10.11. Serre's conjecture in the graded case -- 11. Complexes of free modules. 11.1. McCoy's theorem. 11.2. The rank of a linear map. 11.3. The Eisenbud-Buchsbaum criterion. 11.4. Fitting's ideals. 11.5. The Euler characteristic. 11.6. McRae's invariant. 11.7. The integral character of McRae's invariant

Abelian Groups And Representations Of Finite Partially Ordered Sets

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Author : David Arnold
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-11-14

Abelian Groups And Representations Of Finite Partially Ordered Sets written by David Arnold and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-11-14 with Mathematics categories.

The theme of this book is an exposition of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classification, a description of the objects up to isomorphism, and computation of representation type, a measure of when classification is feasible. David M. Arnold is the Ralph and Jean Storm Professor of Mathematics at Baylor University. He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals.

Algebras Rings And Modules

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Author : Michiel Hazewinkel
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-01-18

Algebras Rings And Modules written by Michiel Hazewinkel and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-18 with Mathematics categories.

Accosiative rings and algebras are very interesting algebraic structures. In a strict sense, the theory of algebras (in particular, noncommutative algebras) originated fromasingleexample,namelythequaternions,createdbySirWilliamR.Hamilton in1843. Thiswasthe?rstexampleofanoncommutative”numbersystem”. During thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative algebras, began to bring some order into them and to single out certain types of algebras for special attention. Thus, low-dimensional algebras, division algebras, and commutative algebras, were classi?ed and characterized. The ?rst complete results in the structure theory of associative algebras over the real and complex ?elds were obtained by T.Molien, E.Cartan and G.Frobenius. Modern ring theory began when J.H.Wedderburn proved his celebrated cl- si?cation theorem for ?nite dimensional semisimple algebras over arbitrary ?elds. Twenty years later, E.Artin proved a structure theorem for rings satisfying both the ascending and descending chain condition which generalized Wedderburn structure theorem. The Wedderburn-Artin theorem has since become a corn- stone of noncommutative ring theory. The purpose of this book is to introduce the subject of the structure theory of associative rings. This book is addressed to a reader who wishes to learn this topic from the beginning to research level. We have tried to write a self-contained book which is intended to be a modern textbook on the structure theory of associative rings and related structures and will be accessible for independent study.

Topics In The Homological Theory Of Modules Over Commutative Rings

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Author : Melvin Hochster
language : en
Publisher: American Mathematical Soc.
Release Date : 1975

Topics In The Homological Theory Of Modules Over Commutative Rings written by Melvin Hochster and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1975 with Mathematics categories.

Contains expository lectures from the CBMS Regional Conference in Mathematics held at the University of Nebraska, June 1974. This book deals mainly with developments and still open questions in the homological theory of modules over commutative (usually, Noetherian) rings.

Modules Over Discrete Valuation Rings

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