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Author: Jinzhong Xu
Publisher: Springer
ISBN: 3540699929
Category : Mathematics
Languages : en
Pages : 167
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Book Description
Since the injective envelope and projective cover were defined by Eckmann and Bas in the 1960s, they have had great influence on the development of homological algebra, ring theory and module theory. In the 1980s, Enochs introduced the flat cover and conjectured that every module has such a cover over any ring. This book provides the uniform methods and systematic treatment to study general envelopes and covers with the emphasis on the existence of flat cover. It shows that Enochs' conjecture is true for a large variety of interesting rings, and then presents the applications of the results. Readers with reasonable knowledge in rings and modules will not have difficulty in reading this book. It is suitable as a reference book and textbook for researchers and graduate students who have an interest in this field.
Author: Jinzhong Xu
Publisher: Springer
ISBN: 3540699929
Category : Mathematics
Languages : en
Pages : 167
Get Book Here
Book Description
Since the injective envelope and projective cover were defined by Eckmann and Bas in the 1960s, they have had great influence on the development of homological algebra, ring theory and module theory. In the 1980s, Enochs introduced the flat cover and conjectured that every module has such a cover over any ring. This book provides the uniform methods and systematic treatment to study general envelopes and covers with the emphasis on the existence of flat cover. It shows that Enochs' conjecture is true for a large variety of interesting rings, and then presents the applications of the results. Readers with reasonable knowledge in rings and modules will not have difficulty in reading this book. It is suitable as a reference book and textbook for researchers and graduate students who have an interest in this field.
Author: J.R. Garcia Rozas
Publisher: CRC Press
ISBN: 9781584880042
Category : Mathematics
Languages : en
Pages : 160
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Book Description
Over the last few years, the study of complexes has become increasingly important. To date, however, most of the research is scattered throughout the literature or available only as lecture notes. Covers and Envelopes in the Category of Complexes of Modules collects these scattered notes and results into a single, concise volume that provides an account of recent developments in the theory and presents several new and important ideas. The author introduces the theory of complexes of modules using only elementary tools-making the field more accessible to non-specialists. He focuses the study on envelopes and covers in this category with respect to some well established and important classes of complexes. He places particular emphasis on DG-injective and DG-projective complexes and flat and DG-flat covers. Other topics covered include Zorn's Lemma for categories, preserving and reflecting covers by functors, orthogonality in the category of complexes, Gorenstein injective and projective complexes, and pure sequences of complexes. Along with its value as a collection of recent work in the field, Covers and Envelopes in the Category of Complexes of Modules presents powerful new ideas that will undoubtedly advance homological methods. Mathematicians-especially researchers in module theory and homological algebra-will welcome this volume as a reference guide and for its new and important results.
Author: J.R. Garcia Rozas
Publisher: Routledge
ISBN: 1351457616
Category : Mathematics
Languages : en
Pages : 152
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Book Description
Over the last few years, the study of complexes has become increasingly important. To date, however, most of the research is scattered throughout the literature or available only as lecture notes. Covers and Envelopes in the Category of Complexes of Modules collects these scattered notes and results into a single, concise volume that provides an account of recent developments in the theory and presents several new and important ideas. The author introduces the theory of complexes of modules using only elementary tools-making the field more accessible to non-specialists. He focuses the study on envelopes and covers in this category with respect to some well established and important classes of complexes. He places particular emphasis on DG-injective and DG-projective complexes and flat and DG-flat covers. Other topics covered include Zorn's Lemma for categories, preserving and reflecting covers by functors, orthogonality in the category of complexes, Gorenstein injective and projective complexes, and pure sequences of complexes. Along with its value as a collection of recent work in the field, Covers and Envelopes in the Category of Complexes of Modules presents powerful new ideas that will undoubtedly advance homological methods. Mathematicians-especially researchers in module theory and homological algebra-will welcome this volume as a reference guide and for its new and important results.
Author: Ashish K. Srivastava
Publisher: Cambridge University Press
ISBN: 1108949533
Category : Mathematics
Languages : en
Pages : 235
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Book Description
Provides a unified treatment of the study of modules invariant under automorphisms of their envelopes and covers.
Author: Pat Goeters
Publisher: CRC Press
ISBN: 142001076X
Category : Mathematics
Languages : en
Pages : 354
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Book Description
About the book In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the par
Author: Rüdiger Göbel
Publisher: Walter de Gruyter
ISBN: 3110218119
Category : Mathematics
Languages : en
Pages : 1002
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Book Description
This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.
Author: T.Y. Lam
Publisher: Springer Science & Business Media
ISBN: 0387488995
Category : Mathematics
Languages : en
Pages : 427
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Book Description
This volume offers a compendium of exercises of varying degree of difficulty in the theory of modules and rings. It is the companion volume to GTM 189. All exercises are solved in full detail. Each section begins with an introduction giving the general background and the theoretical basis for the problems that follow.
Author: Edgar E. Enochs
Publisher: Walter de Gruyter
ISBN: 3110215217
Category : Mathematics
Languages : en
Pages : 377
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Book Description
This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists. In this new edition the authors have added well-known additional material in the first three chapters, and added new material that was not available at the time the original edition was published. In particular, the major changes are the following: Chapter 1: Section 1.2 has been rewritten to clarify basic notions for the beginner, and this has necessitated a new Section 1.3. Chapter 3: The classic work of D. G. Northcott on injective envelopes and inverse polynomials is finally included. This provides additional examples for the reader. Chapter 11: Section 11.9 on Kaplansky classes makes volume one more up to date. The material in this section was not available at the time the first edition was published. The authors also have clarified some text throughout the book and updated the bibliography by adding new references. The book is also suitable for an introductory course in commutative and ordinary homological algebra.
Author: Paul E. Bland
Publisher: Walter de Gruyter
ISBN: 3110250225
Category : Mathematics
Languages : en
Pages : 467
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Book Description
This book is an introduction to the theory of rings and modules that goes beyond what one normally obtains in a graduate course in abstract algebra. In addition to the presentation of standard topics in ring and module theory, it also covers category theory, homological algebra and even more specialized topics like injective envelopes and proj
Author:
Publisher: Elsevier
ISBN: 0080532977
Category : Mathematics
Languages : en
Pages : 1185
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Book Description
Handbook of Algebra
