Graded Ring Theory
Graded Ring Theory

Author: C. Nastasescu

Publisher: Elsevier

Published: 2011-08-18

Total Pages: 352

ISBN-13: 0080960162

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This book is aimed to be a ‘technical’ book on graded rings. By ‘technical’ we mean that the book should supply a kit of tools of quite general applicability, enabling the reader to build up his own further study of non-commutative rings graded by an arbitrary group. The body of the book, Chapter A, contains: categorical properties of graded modules, localization of graded rings and modules, Jacobson radicals of graded rings, the structure thedry for simple objects in the graded sense, chain conditions, Krull dimension of graded modules, homogenization, homological dimension, primary decomposition, and more. One of the advantages of the generality of Chapter A is that it allows direct applications of these results to the theory of group rings, twisted and skew group rings and crossed products. With this in mind we have taken care to point out on several occasions how certain techniques may be specified to the case of strongly graded rings. We tried to write Chapter A in such a way that it becomes suitable for an advanced course in ring theory or general algebra, we strove to make it as selfcontained as possible and we included several problems and exercises. Other chapters may be viewed as an attempt to show how the general techniques of Chapter A can be applied in some particular cases, e.g. the case where the gradation is of type Z. In compiling the material for Chapters B and C we have been guided by our own research interests. Chapter 6 deals with commutative graded rings of type 2 and we focus on two main topics: artihmeticallygraded domains, and secondly, local conditions for Noetherian rings. In Chapter C we derive some structural results relating to the graded properties of the rings considered. The following classes of graded rings receive special attention: fully bounded Noetherian rings, birational extensions of commutative rings, rings satisfying polynomial identities, and Von Neumann regular rings. Here the basic idea is to derive results of ungraded nature from graded information. Some of these sections lead naturally to the study of sheaves over the projective spectrum Proj(R) of a positively graded ring, but we did not go into these topics here. We refer to [125] for a noncommutative treatment of projective geometry, i.e. the geometry of graded P.I. algebras.

Methods of Graded Rings
Methods of Graded Rings

Author: Constantin Nastasescu

Publisher: Springer Science & Business Media

Published: 2004-02-19

Total Pages: 324

ISBN-13: 9783540207467

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The Category of Graded Rings.- The Category of Graded Modules.- Modules over Stronly Graded Rings.- Graded Clifford Theory.- Internal Homogenization.- External Homogenization.- Smash Products.- Localization of Graded Rings.- Application to Gradability.- Appendix A:Some Category Theory.- Appendix B: Dimensions in an abelian Category.- Bibliography.- Index.-

Graded Rings and Graded Grothendieck Groups
Graded Rings and Graded Grothendieck Groups

Author: Roozbeh Hazrat

Publisher: Cambridge University Press

Published: 2016-05-26

Total Pages: 244

ISBN-13: 1316619583

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This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

Algorithms in Invariant Theory
Algorithms in Invariant Theory

Author: Bernd Sturmfels

Publisher: Springer Science & Business Media

Published: 2008-06-17

Total Pages: 202

ISBN-13: 3211774173

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This book is both an easy-to-read textbook for invariant theory and a challenging research monograph that introduces a new approach to the algorithmic side of invariant theory. Students will find the book an easy introduction to this "classical and new" area of mathematics. Researchers in mathematics, symbolic computation, and computer science will get access to research ideas, hints for applications, outlines and details of algorithms, examples and problems.

Rings and Their Modules
Rings and Their Modules

Author: Paul E. Bland

Publisher: Walter de Gruyter

Published: 2011

Total Pages: 467

ISBN-13: 3110250225

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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

Integral Closure of Ideals, Rings, and Modules
Integral Closure of Ideals, Rings, and Modules

Author: Craig Huneke

Publisher: Cambridge University Press

Published: 2006-10-12

Total Pages: 446

ISBN-13: 0521688604

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Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.

Foundations of Module and Ring Theory
Foundations of Module and Ring Theory

Author: Robert Wisbauer

Publisher: Routledge

Published: 2018-05-11

Total Pages: 622

ISBN-13: 1351447343

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This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Ring Constructions and Applications
Ring Constructions and Applications

Author: Andrei V. Kelarev

Publisher: World Scientific

Published: 2002

Total Pages: 218

ISBN-13: 9812799729

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This book contains the definitions of several ring constructions used in various applications. The concept of a groupoid-graded ring includes many of these constructions as special cases and makes it possible to unify the exposition. Recent research results on groupoid-graded rings and more specialized constructions are presented. In addition, there is a chapter containing open problems currently considered in the literature. Ring Constructions and Applications can serve as an excellent introduction for graduate students to many ring constructions as well as to essential basic concepts of group, semigroup and ring theories used in proofs. Contents: Preliminaries; Graded Rings; Examples of Ring Constructions; The Jacobson Radical; Groups of Units; Finiteness Conditions; PI-Rings and Varieties; Gradings of Matrix Rings; Examples of Applications; Open Problems. Readership: Graduate students and researchers using ring constructions in their work.

Graded and Filtered Rings and Modules
Graded and Filtered Rings and Modules

Author: C. Nastasescu

Publisher: Springer

Published: 2006-11-15

Total Pages: 159

ISBN-13: 3540384782

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Anesthesia Student Survival Guide: A Case-Based Approach is an indispensable introduction to the specialty. This concise, easy-to-read, affordable handbook is ideal for medical students, nursing students, and others during the anesthesia rotation. Written in a structured prose format and supplemented with many diagrams, tables, and algorithms, this pocket-sized guide contains essential material covered on the USMLE II-III and other licensing exams. The editors, who are academic faculty at Harvard Medical School, summarize the essential content with 32 informative and compelling case studies designed to help students apply new concepts to real situations. Pharmacology, basic skills, common procedures and anesthesia subspecialties are covered, too, with just the right amount of detail for an introductory text. The unique book also offers a section containing career advice and insider tips on how to receive good evaluations from supervising physicians. With its combination of astute clinical instruction, basic science explanation, and practical tips from physicians that have been there before, this handbook is your one-stop guide to a successful anesthesia rotation.

Leavitt Path Algebras
Leavitt Path Algebras

Author: Gene Abrams

Publisher: Springer

Published: 2017-11-30

Total Pages: 296

ISBN-13: 1447173449

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This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and symbolic dynamics. With its descriptive writing style, this book is highly accessible.