Extensions Of Rings And Modules

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Extensions of Rings and Modules

Gary F. Birkenmeier,Jae Keol Park,S Tariq Rizvi
Extensions of Rings and Modules Book PDF

Author : Gary F. Birkenmeier,Jae Keol Park,S Tariq Rizvi
Publisher : Springer Science & Business Media
Page : 442 pages
File Size : 54,6 Mb
Release : 2013-07-19
Category : Mathematics
ISBN : 9780387927169

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Extensions of Rings and Modules by Gary F. Birkenmeier,Jae Keol Park,S Tariq Rizvi Pdf

The "extensions" of rings and modules have yet to be explored in detail in a research monograph. This book presents state of the art research and also stimulating new and further research. Broken into three parts, Part I begins with basic notions, terminology, definitions and a description of the classes of rings and modules. Part II considers the transference of conditions between a base ring or module and its extensions. And Part III utilizes the concept of a minimal essental extension with respect to a specific class (a hull). Mathematical interdisciplinary applications appear throughout. Major applications of the ring and module theory to Functional Analysis, especially C*-algebras, appear in Part III, make this book of interest to Algebra and Functional Analysis researchers. Notes and exercises at the end of every chapter, and open problems at the end of all three parts, lend this as an ideal textbook for graduate or advanced undergradate students.

Extensions of Rings and Modules

Gary F. Birkenmeier,Jae Keol Park,SYED T RIZVI
Extensions of Rings and Modules Book PDF

Author : Gary F. Birkenmeier,Jae Keol Park,SYED T RIZVI
Publisher : Springer
Page : 432 pages
File Size : 53,6 Mb
Release : 2013-09-14
Category : Mathematics
ISBN : 0387927573

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Extensions of Rings and Modules by Gary F. Birkenmeier,Jae Keol Park,SYED T RIZVI Pdf

The "extensions" of rings and modules have yet to be explored in detail in a research monograph. This book presents state of the art research and also stimulating new and further research. Broken into three parts, Part I begins with basic notions, terminology, definitions and a description of the classes of rings and modules. Part II considers the transference of conditions between a base ring or module and its extensions. And Part III utilizes the concept of a minimal essental extension with respect to a specific class (a hull). Mathematical interdisciplinary applications appear throughout. Major applications of the ring and module theory to Functional Analysis, especially C*-algebras, appear in Part III, make this book of interest to Algebra and Functional Analysis researchers. Notes and exercises at the end of every chapter, and open problems at the end of all three parts, lend this as an ideal textbook for graduate or advanced undergradate students.

Algebras, Rings and Modules

Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko
Algebras Rings and Modules Book PDF

Author : Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko
Publisher : Springer Science & Business Media
Page : 393 pages
File Size : 47,9 Mb
Release : 2006-01-18
Category : Mathematics
ISBN : 9781402026911

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Algebras, Rings and Modules by Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko Pdf

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.

Rings, Modules, and the Total

Friedrich Kasch,Adolf Mader
Rings Modules and the Total Book PDF

Author : Friedrich Kasch,Adolf Mader
Publisher : Springer Science & Business Media
Page : 154 pages
File Size : 52,8 Mb
Release : 2004-06-25
Category : Mathematics
ISBN : 3764371250

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Rings, Modules, and the Total by Friedrich Kasch,Adolf Mader Pdf

Accessible to anyone with a basic knowledge of ring and module theory A short introduction to torsion-free Abelian groups is included

Lectures on Rings and Modules

Joachim Lambek
Lectures on Rings and Modules Book PDF

Author : Joachim Lambek
Publisher : American Mathematical Soc.
Page : 196 pages
File Size : 55,5 Mb
Release : 2009
Category : Associative rings
ISBN : 9780821849002

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Lectures on Rings and Modules by Joachim Lambek Pdf

This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a particular emphasis on the concept of the complete ring of quotients. A survey of the fundamental concepts of algebras in the first chapter helps to make the treatment self-contained. The topics covered include selected results on Boolean and other commutative rings, the classical structure theory of associative rings, injective modules, and rings of quotients. The final chapter provides an introduction to homological algebra. Besides three appendices on further results, there is a six-page section of historical comments. Table of Contents: Fundamental Concepts of Algebra: 1.1 Rings and related algebraic systems; 1.2 Subrings, homomorphisms, ideals; 1.3 Modules, direct products, and direct sums; 1.4 Classical isomorphism theorems. Selected Topics on Commutative Rings: 2.1 Prime ideals in commutative rings; 2.2 Prime ideals in special commutative rings; 2.3 The complete ring of quotients of a commutative ring; 2.4 Rings of quotients of commutative semiprime rings; 2.5 Prime ideal spaces.Classical Theory of Associative Rings: 3.1 Primitive rings; 3.2 Radicals; 3.3 Completely reducible modules; 3.4 Completely reducible rings; 3.5 Artinian and Noetherian rings; 3.6 On lifting idempotents; 3.7 Local and semiperfect rings. Injectivity and Related Concepts: 4.1 Projective modules; 4.2 Injective modules; 4.3 The complete ring of quotients; 4.4 Rings of endomorphisms of injective modules; 4.5 Regular rings of quotients; 4.6 Classical rings of quotients; 4.7 The Faith-Utumi theorem. Introduction to Homological Algebra: 5.1 Tensor products of modules; 5.2 Hom and $\otimes$ as functors; 5.3 Exact sequences; 5.4 Flat modules; 5.5 Torsion and extension products. Appendixes; Comments; Bibliography; Index. Review from Zentralblatt Math: Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. Review from American Mathematical Monthly: An introduction to associative rings and modules which requires of the reader only the mathematical maturity which one would attain in a first-year graduate algebra [course]...in order to make the contents of the book as accessible as possible, the author develops all the fundamentals he will need.In addition to covering the basic topics...the author covers some topics not so readily available to the nonspecialist...the chapters are written to be as independent as possible...[which will be appreciated by] students making their first acquaintance with the subject...one of the most successful features of the book is that it can be read by graduate students with little or no help from a specialist. (CHEL/283.H)

Modules and Rings

John Dauns
Modules and Rings Book PDF

Author : John Dauns
Publisher : Cambridge University Press
Page : 470 pages
File Size : 50,8 Mb
Release : 1994-10-28
Category : Mathematics
ISBN : 9780521462587

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Modules and Rings by John Dauns Pdf

This book on modern module and non-commutative ring theory is ideal for beginning graduate students. It starts at the foundations of the subject and progresses rapidly through the basic concepts to help the reader reach current research frontiers. Students will have the chance to develop proofs, solve problems, and to find interesting questions. The first half of the book is concerned with free, projective, and injective modules, tensor algebras, simple modules and primitive rings, the Jacobson radical, and subdirect products. Later in the book, more advanced topics, such as hereditary rings, categories and functors, flat modules, and purity are introduced. These later chapters will also prove a useful reference for researchers in non-commutative ring theory. Enough background material (including detailed proofs) is supplied to give the student a firm grounding in the subject.

Integral Closure of Ideals, Rings, and Modules

Craig Huneke,Irena Swanson
Integral Closure of Ideals Rings and Modules Book PDF

Author : Craig Huneke,Irena Swanson
Publisher : Cambridge University Press
Page : 446 pages
File Size : 48,5 Mb
Release : 2006-10-12
Category : Mathematics
ISBN : 9780521688604

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Integral Closure of Ideals, Rings, and Modules by Craig Huneke,Irena Swanson Pdf

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

Skew PBW Extensions

William Fajardo,Claudia Gallego,Oswaldo Lezama,Armando Reyes,Héctor Suárez,Helbert Venegas
Skew PBW Extensions Book PDF

Author : William Fajardo,Claudia Gallego,Oswaldo Lezama,Armando Reyes,Héctor Suárez,Helbert Venegas
Publisher : Springer
Page : 584 pages
File Size : 54,6 Mb
Release : 2021-12-13
Category : Mathematics
ISBN : 3030533808

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Skew PBW Extensions by William Fajardo,Claudia Gallego,Oswaldo Lezama,Armando Reyes,Héctor Suárez,Helbert Venegas Pdf

This monograph is devoted to a new class of non-commutative rings, skew Poincaré–Birkhoff–Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Gröbner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin–Schelter regular algebras, and the noncommutative Zariski cancellation problem. The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.

Lattice-ordered Rings and Modules

Stuart A. Steinberg
Lattice ordered Rings and Modules Book PDF

Author : Stuart A. Steinberg
Publisher : Springer Science & Business Media
Page : 639 pages
File Size : 42,5 Mb
Release : 2009-11-19
Category : Mathematics
ISBN : 9781441917218

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Lattice-ordered Rings and Modules by Stuart A. Steinberg Pdf

This book provides an exposition of the algebraic aspects of the theory of lattice-ordered rings and lattice-ordered modules. All of the background material on rings, modules, and lattice-ordered groups necessary to make the work self-contained and accessible to a variety of readers is included. Filling a gap in the literature, Lattice-Ordered Rings and Modules may be used as a textbook or for self-study by graduate students and researchers studying lattice-ordered rings and lattice-ordered modules. Steinberg presents the material through 800+ extensive examples of varying levels of difficulty along with numerous exercises at the end of each section. Key topics include: lattice-ordered groups, rings, and fields; archimedean $l$-groups; f-rings and larger varieties of $l$-rings; the category of f-modules; various commutativity results.

Rings, Polynomials, and Modules

Marco Fontana,Sophie Frisch,Sarah Glaz,Francesca Tartarone,Paolo Zanardo
Rings Polynomials and Modules Book PDF

Author : Marco Fontana,Sophie Frisch,Sarah Glaz,Francesca Tartarone,Paolo Zanardo
Publisher : Springer
Page : 375 pages
File Size : 46,7 Mb
Release : 2017-11-11
Category : Mathematics
ISBN : 9783319658742

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Rings, Polynomials, and Modules by Marco Fontana,Sophie Frisch,Sarah Glaz,Francesca Tartarone,Paolo Zanardo Pdf

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.

Foundations of Module and Ring Theory

Robert Wisbauer
Foundations of Module and Ring Theory Book PDF

Author : Robert Wisbauer
Publisher : Routledge
Page : 425 pages
File Size : 51,9 Mb
Release : 2018-05-11
Category : Mathematics
ISBN : 9781351447348

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Foundations of Module and Ring Theory by Robert Wisbauer Pdf

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.

Advances in Rings and Modules

Sergio R. López-Permouth,Jae Keol Park,S. Tariq Rizvi,Cosmin S. Roman
Advances in Rings and Modules Book PDF

Author : Sergio R. López-Permouth,Jae Keol Park,S. Tariq Rizvi,Cosmin S. Roman
Publisher : American Mathematical Soc.
Page : 283 pages
File Size : 46,6 Mb
Release : 2018-09-06
Category : Modules (Algebra)
ISBN : 9781470435554

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Advances in Rings and Modules by Sergio R. López-Permouth,Jae Keol Park,S. Tariq Rizvi,Cosmin S. Roman Pdf

This volume, dedicated to Bruno J. Müller, a renowned algebraist, is a collection of papers that provide a snapshot of the diversity of themes and applications that interest algebraists today. The papers highlight the latest progress in ring and module research and present work done on the frontiers of the topics discussed. In addition, selected expository articles are included to give algebraists and other mathematicians, including graduate students, an accessible introduction to areas that may be outside their own expertise.

Ring and Module Theory

Toma Albu,Gary F. Birkenmeier,Ali Erdogan,Adnan Tercan
Ring and Module Theory Book PDF

Author : Toma Albu,Gary F. Birkenmeier,Ali Erdogan,Adnan Tercan
Publisher : Springer Science & Business Media
Page : 200 pages
File Size : 53,9 Mb
Release : 2011-02-04
Category : Mathematics
ISBN : 9783034600071

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Ring and Module Theory by Toma Albu,Gary F. Birkenmeier,Ali Erdogan,Adnan Tercan Pdf

This book is a collection of invited papers and articles, many presented at the 2008 International Conference on Ring and Module Theory. The papers explore the latest in various areas of algebra, including ring theory, module theory and commutative algebra.

Algebras, Rings and Modules, Volume 2

Michiel Hazewinkel,Nadiya M. Gubareni
Algebras Rings and Modules Volume 2 Book PDF

Author : Michiel Hazewinkel,Nadiya M. Gubareni
Publisher : CRC Press
Page : 364 pages
File Size : 50,9 Mb
Release : 2017-04-11
Category : Mathematics
ISBN : 9781351869874

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Algebras, Rings and Modules, Volume 2 by Michiel Hazewinkel,Nadiya M. Gubareni Pdf

The theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This is the second volume of Algebras, Rings and Modules: Non-commutative Algebras and Rings by M. Hazewinkel and N. Gubarenis, a continuation stressing the more important recent results on advanced topics of the structural theory of associative algebras, rings and modules.

Rings and Their Modules

Paul E. Bland
Rings and Their Modules Book PDF

Author : Paul E. Bland
Publisher : Walter de Gruyter
Page : 467 pages
File Size : 41,6 Mb
Release : 2011-02-01
Category : Mathematics
ISBN : 9783110250237

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Rings and Their Modules by Paul E. Bland Pdf

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. The theme of the text is the interplay between rings and modules. At times rings are investigated by considering given sets of conditions on the modules they admit and at other times rings of a certain type are considered to see what structure is forced on their modules. Standard topics in ring and module theory such as chain conditions on rings and modules, injective and projective modules and semisimple rings are included as well as subjects like category theory and homological algebra. The text also contains presentations on topics such as flat modules and coherent rings, injective envelopes, projective covers and perfect rings, reflexive modules and quasi-Frobenius rings, and graded rings and modules. The book is a self-contained volume written in a very systematic style: all proofs are clear and easy for the reader to understand and all arguments are based on materials contained in the book. A problem sets follow each section. It is assumed that the reader is familiar with concepts such as Zorn's lemma, commutative diagrams and ordinal and cardinal numbers. It is also assumed that the reader has a basic knowledge of rings and their homomorphisms. The text is suitable for graduate and PhD students who have chosen ring theory for their research subject.