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| module theory books: Module Theory Thomas Scott Blyth, 1990 This textbook provides a self-contained course on the basic properties of modules and their importance in the theory of linear algebra. The first 11 chapters introduce the central results and applications of the theory of modules. Subsequent chapters deal with advanced linear algebra, including multilinear and tensor algebra, and explore such topics as the exterior product approach to the determinants of matrices, a module-theoretic approach to the structure of finitely generated Abelian groups, canonical forms, and normal transformations. Suitable for undergraduate courses, the text now includes a proof of the celebrated Wedderburn-Artin theorem which determines the structure of simple Artinian rings. |
| module theory books: A First Course In Module Theory Mike E Keating, 1998-07-31 This book is an introduction to module theory for the reader who knows something about linear algebra and ring theory. Its main aim is the derivation of the structure theory of modules over Euclidean domains. This theory is applied to obtain the structure of abelian groups and the rational canonical and Jordan normal forms of matrices. The basic facts about rings and modules are given in full generality, so that some further topics can be discussed, including projective modules and the connection between modules and representations of groups.The book is intended to serve as supplementary reading for the third or fourth year undergraduate who is taking a course in module theory. The further topics point the way to some projects that might be attempted in conjunction with a taught course. |
| module theory books: Module Theory Alberto Facchini, 2012-02-03 This book presents topics in module theory and ring theory: some, such as Goldie dimension and semiperfect rings are now considered classical and others more specialized, such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules. |
| module theory books: Algebra William A. Adkins, Steven H. Weintraub, 1992-09-03 First year graduate algebra text. The choice of topics is guided by the underlying theme of modules as a basic unifying concept in mathematics. Beginning with standard topics in group and ring theory, the authors then develop basic module theory and its use in investigating bilinear, sesquilinear, and quadratic forms. Annotation copyrighted by Book News, Inc., Portland, OR |
| module theory books: A First Course in Module Theory M. E. Keating, 1998-01-01 An introduction to module theory for students with some knowledge of linear algebra and elementary ring theory. Expounds the basics of module theory, including methods of comparing, constructing and decomposing modules, then presents the structure theory of modules over Euclidean domains. Concluding chapters look at two standard forms for a square matrix, and projective modules over rings in general. Annotation copyrighted by Book News, Inc., Portland, OR |
| module theory books: Ring and Module Theory Toma Albu, Gary F. Birkenmeier, Ali Erdogan, Adnan Tercan, 2011-02-04 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. |
| module theory books: Lectures on Modules and Rings Tsit-Yuen Lam, 1999 This new book can be read independently from the first volume and may be used for lecturing, seminar- and self-study, or for general reference. It focuses more on specific topics in order to introduce readers to a wealth of basic and useful ideas without the hindrance of heavy machinery or undue abstractions. User-friendly with its abundance of examples illustrating the theory at virtually every step, the volume contains a large number of carefully chosen exercises to provide newcomers with practice, while offering a rich additional source of information to experts. A direct approach is used in order to present the material in an efficient and economic way, thereby introducing readers to a considerable amount of interesting ring theory without being dragged through endless preparatory material. |
| module theory books: Foundations of Module and Ring Theory Robert Wisbauer, 2018-05-11 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. |
| module theory books: Module Theory, Extending Modules and Generalizations Adnan Tercan, Canan C. Yücel, 2016-05-13 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. |
| module theory books: Exercises in Modules and Rings T.Y. Lam, 2009-12-08 The idea of writing this book came roughly at the time of publication of my graduate text Lectures on Modules and Rings, Springer GTM Vol. 189, 1999. Since that time, teaching obligations and intermittent intervention of other projects caused prolonged delays in the work on this volume. Only a lucky break in my schedule in 2006 enabled me to put the finishing touches on the completion of this long overdue book. This book is intended to serve a dual purpose. First, it is designed as a problem book for Lectures. As such, it contains the statements and full solutions of the many exercises that appeared in Lectures. Second, this book is also offered as a reference and repository for general information in the theory of modules and rings that may be hard to find in the standard textbooks in the field. As a companion volume to Lectures, this work covers the same math ematical material as its parent work; namely, the part of ring theory that makes substantial use of the notion of modules. The two books thus share the same table of contents, with the first half treating projective, injective, and flat modules, homological and uniform dimensions, and the second half dealing with noncommutative localizations and Goldie's theorems, maximal rings of quotients, Frobenius and quasi-Frobenius rings, conclud ing with Morita's theory of category equivalences and dualities. |
| module theory books: Rings and Categories of Modules Frank W. Anderson, Kent R. Fuller, 2012-12-06 This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semi perfect and perfect rings. In this second edition we have included a chapter containing many of the classical results on artinian rings that have hdped to form the foundation for much of the contemporary research on the representation theory of artinian rings and finite dimensional algebras. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course many important areas of ring and module theory that the text does not touch upon. |
| module theory books: Lattice Concepts of Module Theory Grigore Calugareanu, 2013-04-17 It became more and more usual, from, say, the 1970s, for each book on Module Theory, to point out and prove some (but in no more than 15 to 20 pages) generalizations to (mostly modular) lattices. This was justified by the nowadays widely accepted perception that the structure of a module over a ring is best understood in terms of the lattice struc ture of its submodule lattice. Citing Louis H. Rowen this important example (the lattice of all the submodules of a module) is the raison d'etre for the study of lattice theory by ring theorists. Indeed, many module-theoretic results can be proved by using lattice theory alone. The purpose of this book is to collect and present all and only the results of this kind, although for this purpose one must develop some significant lattice theory. The results in this book are of the following categories: the folklore of Lattice Theory (to be found in each Lattice Theory book), module theoretic results generalized in (modular, and possibly compactly gen erated) lattices (to be found in some 6 to 7 books published in the last 20 years), very special module-theoretic results generalized in lattices (e. g. , purity in Chapter 9 and several dimensions in Chapter 13, to be found mostly in [27], respectively, [34] and [18]) and some new con cepts (e. g. |
| module theory books: Module Theory Thomas Scott Blyth, 1990 This textbook provides a self-contained course on the basic properties of modules and their importance in the theory of linear algebra. The first 11 chapters introduce the central results and applications of the theory of modules. Subsequent chapters deal with advanced linear algebra, including multilinear and tensor algebra, and explore such topics as the exterior product approach to the determinants of matrices, a module-theoretic approach to the structure of finitely generated Abelian groups, canonical forms, and normal transformations. Suitable for undergraduate courses, the text now includes a proof of the celebrated Wedderburn-Artin theorem which determines the structure of simple Artinian rings. |
| module theory books: Stable Module Theory Maurice Auslander, Mark Bridger, 1969 The notions of torsion and torsion freeness have played a very important role in module theory--particularly in the study of modules over integral domains. Furthermore, the use of homological techniques in this connection has been well established. It is the aim of this paper to extend these techniques and to show that this extension leads naturally to several new concepts (e.g. k-torsion freeness and Gorenstein dimension) which are useful in the classification of modules and rings. |
| module theory books: Rings and Their Modules Paul E. Bland, 2011 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 |
| module theory books: Fields and Rings Irving Kaplansky, 1972 This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules. In all three parts of this book the author lives up to his reputation as a first-rate mathematical stylist. Throughout the work the clarity and precision of the presentation is not only a source of constant pleasure but will enable the neophyte to master the material here presented with dispatch and ease.—A. Rosenberg, Mathematical Reviews |
| module theory books: Lectures on Modules and Rings Tsit-Yuen Lam, 2012-12-06 Textbook writing must be one of the cruelest of self-inflicted tortures. - Carl Faith Math Reviews 54: 5281 So why didn't I heed the warning of a wise colleague, especially one who is a great expert in the subject of modules and rings? The answer is simple: I did not learn about it until it was too late! My writing project in ring theory started in 1983 after I taught a year-long course in the subject at Berkeley. My original plan was to write up my lectures and publish them as a graduate text in a couple of years. My hopes of carrying out this plan on schedule were, however, quickly dashed as I began to realize how much material was at hand and how little time I had at my disposal. As the years went by, I added further material to my notes, and used them to teach different versions of the course. Eventually, I came to the realization that writing a single volume would not fully accomplish my original goal of giving a comprehensive treatment of basic ring theory. At the suggestion of Ulrike Schmickler-Hirzebruch, then Mathematics Editor of Springer-Verlag, I completed the first part of my project and published the write up in 1991 as A First Course in Noncommutative Rings, GTM 131, hereafter referred to as First Course (or simply FC). |
| module theory books: Introduction to Liaison Theory and Deficiency Modules Juan C. Migliore, 2012-12-06 In the fall of 1992 I was invited by Professor Changho Keem to visit Seoul National University and give a series of talks. I was asked to write a monograph based on my talks, and the result was published by the Global Analysis Research Center of that University in 1994. The monograph treated deficiency modules and liaison theory for complete intersections. Over the next several years I continually thought of improvements and additions that I would like to make to the manuscript, and at the same time my research led me in directions that gave me a fresh perspective on much of the material, especially in the direction of liaison theory. This re sulted in a dramatic change in the focus of this manuscript, from complete intersections to Gorenstein ideals, and a substantial amount of additions and revisions. It is my hope that this book now serves not only as an introduction to a beautiful subject, but also gives the reader a glimpse at very recent developments and an idea of the direction in which liaison theory is going, at least from my perspective. One theme which I have tried to stress is the tremendous amount of geometry which lies at the heart of the subject, and the beautiful interplay between algebra and geometry. Whenever possible I have given remarks and examples to illustrate this interplay, and I have tried to phrase the results in as geometric a way as possible. |
| module theory books: Algebra I. Martin Isaacs, 2009 as a student. --Book Jacket. |
| module theory books: Algebraic D-modules Armand Borel, 1987 Presented here are recent developments in the algebraic theory of D-modules. The book contains an exposition of the basic notions and operations of D-modules, of special features of coherent, holonomic, and regular holonomic D-modules, and of the Riemann-Hilbert correspondence. The theory of Algebraic D-modules has found remarkable applications outside of analysis proper, in particular to infinite dimensional representations of semisimple Lie groups, to representations of Weyl groups, and to algebraic geometry. |
| module theory books: Foundations of Commutative Rings and Their Modules Fanggui Wang, Hwankoo Kim, 2017-01-06 This book provides an introduction to the basics and recent developments of commutative algebra. A glance at the contents of the first five chapters shows that the topics covered are ones that usually are included in any commutative algebra text. However, the contents of this book differ significantly from most commutative algebra texts: namely, its treatment of the Dedekind–Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings and the valuative dimension, and Nagata rings. Going further, Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of the pullbacks, especially Milnor squares and D+M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings with finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated on by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass–Quillen problem is discussed. Finally, Chapter 11 aims to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules. Each section of the book is followed by a selection of exercises of varying degrees of difficulty. This book will appeal to a wide readership from graduate students to academic researchers who are interested in studying commutative algebra. |
| module theory books: Representation Theory of Finite Groups Benjamin Steinberg, 2011-10-23 This book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics. |
| module theory books: Basic Algebra I Nathan Jacobson, 2012-12-11 A classic text and standard reference for a generation, this volume covers all undergraduate algebra topics, including groups, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. 1985 edition. |
| module theory books: Ring Theory Kenneth Goodearl, 1976-03-01 |
| module theory books: Extending Modules Nguyen Viet Dung, 2019-01-22 Module theory is an important tool for many different branches of mathematics, as well as being an interesting subject in its own right. Within module theory, the concept of injective modules is particularly important. Extending modules form a natural class of modules which is more general than the class of injective modules but retains many of its |
| module theory books: Visual Group Theory Nathan Carter, 2021-06-08 Recipient of the Mathematical Association of America's Beckenbach Book Prize in 2012! Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts, but its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective. The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. |
| module theory books: Steps in Commutative Algebra R. Y. Sharp, 2000 Introductory account of commutative algebra, aimed at students with a background in basic algebra. |
| module theory books: Lectures on Rings and Modules Joachim Lambek, 1966 |
| module theory books: Basic Representation Theory of Algebras Ibrahim Assem, Flávio U. Coelho, 2020-04-03 This textbook introduces the representation theory of algebras by focusing on two of its most important aspects: the Auslander–Reiten theory and the study of the radical of a module category. It starts by introducing and describing several characterisations of the radical of a module category, then presents the central concepts of irreducible morphisms and almost split sequences, before providing the definition of the Auslander–Reiten quiver, which encodes much of the information on the module category. It then turns to the study of endomorphism algebras, leading on one hand to the definition of the Auslander algebra and on the other to tilting theory. The book ends with selected properties of representation-finite algebras, which are now the best understood class of algebras. Intended for graduate students in representation theory, this book is also of interest to any mathematician wanting to learn the fundamentals of this rapidly growing field. A graduate course in non-commutative or homological algebra, which is standard in most universities, is a prerequisite for readers of this book. |
| module theory books: Lessons on Rings, Modules and Multiplicities D. G. Northcott, 2008-12-11 This volume provides a clear and self-contained introduction to important results in the theory of rings and modules. Assuming only the mathematical background provided by a normal undergraduate curriculum, the theory is derived by comparatively direct and simple methods. It will be useful to both undergraduates and research students specialising in algebra. In his usual lucid style the author introduces the reader to advanced topics in a manner which makes them both interesting and easy to assimilate. As the text gives very full explanations, a number of well-ordered exercises are included at the end of each chapter. These lead on to further significant results and give the reader an opportunity to devise his own arguments and to test his understanding of the subject. |
| module theory books: Extensions of Rings and Modules Gary F. Birkenmeier, Jae Keol Park, S Tariq Rizvi, 2013-07-19 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. |
| module theory books: Algebras, Rings and Modules Michiel Hazewinkel, Nadiya M. Gubareni, 2016-04-05 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 centu |
| module theory books: Advanced Modern Algebra Joseph J. Rotman, 2023-02-22 This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study. |
| module theory books: Linear Algebra over Commutative Rings Bernard R. McDonald, 2020-11-26 This monograph arose from lectures at the University of Oklahoma on topics related to linear algebra over commutative rings. It provides an introduction of matrix theory over commutative rings. The monograph discusses the structure theory of a projective module. |
| module theory books: Unstable Modules Over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture Lionel Schwartz, 1994-07-15 A comprehensive account of one of the main directions of algebraic topology, this book focuses on the Sullivan conjecture and its generalizations and applications. Lionel Schwartz collects here for the first time some of the most innovative work on the theory of modules over the Steenrod algebra, including ideas on the Segal conjecture, work from the late 1970s by Adams and Wilkerson, and topics in algebraic group representation theory. This course-tested book provides a valuable reference for algebraic topologists and includes foundational material essential for graduate study. |
| module theory books: An Introduction to Hopf Algebras Robert G. Underwood, 2011-08-30 Only book on Hopf algebras aimed at advanced undergraduates |
| module theory books: Module Theory C. Faith, S. Wiegand, 2014-01-15 |
| module theory books: Module Theory C. Faith, S. Wiegand, 2006-11-15 |
| module theory books: Exercises in Classical Ring Theory T.Y. Lam, 2013-06-29 Based in large part on the comprehensive First Course in Ring Theory by the same author, this book provides a comprehensive set of problems and solutions in ring theory that will serve not only as a teaching aid to instructors using that book, but also for students, who will see how ring theory theorems are applied to solving ring-theoretic problems and how good proofs are written. The author demonstrates that problem-solving is a lively process: in Comments following many solutions he discusses what happens if a hypothesis is removed, whether the exercise can be further generalized, what would be a concrete example for the exercise, and so forth. The book is thus much more than a solution manual. |
MODULE Definition & Meaning - Merriam-Webster
The meaning of MODULE is a standard or unit of measurement. How to use module in a sentence.
MODULE | English meaning - Cambridge Dictionary
MODULE definition: 1. one of a set of separate parts that, when combined, form a complete whole: 2. one of the units…. Learn more.
Module - definition of module by The Free Dictionary
1. a separable component, frequently one that is interchangeable with others, for assembly into units of differing size, complexity, or function. 2. any of the self-contained segments of a …
module noun - Definition, pictures, pronunciation and usage ...
Definition of module noun from the Oxford Advanced Learner's Dictionary. a unit that can form part of a course of study, especially at a college or university in the UK. The course consists of …
MODULE definition and meaning | Collins English Dictionary
A module is a part of a machine, especially a computer, which performs a particular function.
Modules (since C++20) - cppreference.com
Sep 13, 2024 · If any identifier in the module name or module partition is defined as an object-like macro, the program is ill-formed. A named module is the collection of module units with the …
6. Modules — Python 3.13.5 documentation
2 days ago · A module is a file containing Python definitions and statements. The file name is the module name with the suffix .py appended. Within a module, the module’s name (as a string) …
MODULE Definition & Meaning - Merriam-Webster
The meaning of MODULE is a standard or unit of measurement. How to use module in a sentence.
MODULE | English meaning - Cambridge Dictionary
MODULE definition: 1. one of a set of separate parts that, when combined, form a complete whole: 2. one of the units…. Learn more.
Module - definition of module by The Free Dictionary
1. a separable component, frequently one that is interchangeable with others, for assembly into units of differing size, complexity, or function. 2. any of the self-contained segments of a …
module noun - Definition, pictures, pronunciation and usage ...
Definition of module noun from the Oxford Advanced Learner's Dictionary. a unit that can form part of a course of study, especially at a college or university in the UK. The course consists of …
MODULE definition and meaning | Collins English Dictionary
A module is a part of a machine, especially a computer, which performs a particular function.
Modules (since C++20) - cppreference.com
Sep 13, 2024 · If any identifier in the module name or module partition is defined as an object-like macro, the program is ill-formed. A named module is the collection of module units with the …
6. Modules — Python 3.13.5 documentation
2 days ago · A module is a file containing Python definitions and statements. The file name is the module name with the suffix .py appended. Within a module, the module’s name (as a string) …
