# topics in commutative ring theory

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## Topics In Commutative Ring Theory

**Author :**John J. Watkins

**ISBN :**9781400828173

**Genre :**Mathematics

**File Size :**65. 55 MB

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Topics in Commutative Ring Theory is a textbook for advanced undergraduate students as well as graduate students and mathematicians seeking an accessible introduction to this fascinating area of abstract algebra. Commutative ring theory arose more than a century ago to address questions in geometry and number theory. A commutative ring is a set-such as the integers, complex numbers, or polynomials with real coefficients--with two operations, addition and multiplication. Starting from this simple definition, John Watkins guides readers from basic concepts to Noetherian rings-one of the most important classes of commutative rings--and beyond to the frontiers of current research in the field. Each chapter includes problems that encourage active reading--routine exercises as well as problems that build technical skills and reinforce new concepts. The final chapter is devoted to new computational techniques now available through computers. Careful to avoid intimidating theorems and proofs whenever possible, Watkins emphasizes the historical roots of the subject, like the role of commutative rings in Fermat's last theorem. He leads readers into unexpected territory with discussions on rings of continuous functions and the set-theoretic foundations of mathematics. Written by an award-winning teacher, this is the first introductory textbook to require no prior knowledge of ring theory to get started. Refreshingly informal without ever sacrificing mathematical rigor, Topics in Commutative Ring Theory is an ideal resource for anyone seeking entry into this stimulating field of study

## Topics In Commutative Ring Theory

**Author :**Irving Kaplansky

**ISBN :**UOM:39015017316541

**Genre :**Mathematics

**File Size :**73. 31 MB

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## Commutative Ring Theory

**Author :**H. Matsumura

**ISBN :**0521367646

**Genre :**Mathematics

**File Size :**77. 79 MB

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This book explores commutative ring theory, an important a foundation for algebraic geometry and complex analytical geometry.

## Non Noetherian Commutative Ring Theory

**Author :**S.T. Chapman

**ISBN :**9781475731804

**Genre :**Mathematics

**File Size :**63. 20 MB

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Commutative Ring Theory emerged as a distinct field of research in math ematics only at the beginning of the twentieth century. It is rooted in nine teenth century major works in Number Theory and Algebraic Geometry for which it provided a useful tool for proving results. From this humble origin, it flourished into a field of study in its own right of an astonishing richness and interest. Nowadays, one has to specialize in an area of this vast field in order to be able to master its wealth of results and come up with worthwhile contributions. One of the major areas of the field of Commutative Ring Theory is the study of non-Noetherian rings. The last ten years have seen a lively flurry of activity in this area, including: a large number of conferences and special sections at national and international meetings dedicated to presenting its results, an abundance of articles in scientific journals, and a substantial number of books capturing some of its topics. This rapid growth, and the occasion of the new Millennium, prompted us to embark on a project aimed at presenting an overview of the recent research in the area. With this in mind, we invited many of the most prominent researchers in Non-Noetherian Commutative Ring Theory to write expository articles representing the most recent topics of research in this area.

## Topics In The Homological Theory Of Modules Over Commutative Rings

**Author :**Melvin Hochster

**ISBN :**0821888714

**Genre :**Mathematics

**File Size :**79. 21 MB

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This volume contains expository lectures by Melvin Hochster from the CBMS Regional Conference in Mathematics held at the University of Nebraska, June 1974. The lectures deal mainly with recent developments and still open questions in the homological theory of modules over commutative (usually, Noetherian) rings. A good deal of attention is given to the role ``big'' Cohen-Macaulay modules play in clearing up some of the open questions. A modest knowledge of commutative rings and familarity with (the long exact sequences for) Tor and Ext should suffice as a background for the reader.

## Advances In Commutative Ring Theory

**Author :**David Dobbs

**ISBN :**0824771478

**Genre :**Mathematics

**File Size :**86. 18 MB

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"Presents the proceedings of the recently held Third International Conference on Commutative Ring Theory in Fez, Morocco. Details the latest developments in commutative algebra and related areas-featuring 26 original research articles and six survey articles on fundamental topics of current interest. Examines wide-ranging developments in commutative algebra, together with connections to algebraic number theory and algebraic geometry."

## Undergraduate Commutative Algebra

**Author :**Miles Reid

**ISBN :**0521458897

**Genre :**Mathematics

**File Size :**40. 26 MB

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In this well-written introduction to commutative algebra, the author shows the link between commutative ring theory and algebraic geometry. In addition to standard material, the book contrasts the methods and ideology of modern abstract algebra with concrete applications in algebraic geometry and number theory. Professor Reid begins with a discussion of modules and Noetherian rings before moving on to finite extensions and the Noether normalization. Sections on the nullstellensatz and rings of fractions precede sections on primary decomposition and normal integral domains. This book is ideal for anyone seeking a primer on commutative algebra.

## Commutative Ring Theory And Applications

**Author :**Marco Fontana

**ISBN :**0203910621

**Genre :**Mathematics

**File Size :**65. 34 MB

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Featuring presentations from the Fourth International Conference on Commutative Algebra held in Fez, Morocco, this reference presents trends in the growing area of commutative algebra. With contributions from nearly 50 internationally renowned researchers, the book emphasizes innovative applications and connections to algebraic number theory, geometry, and homological and computational algebra. Presenting challenging problems of contemporary interest, discussions include linear Diophantine equations, going-down and going-up properties, and graded modules and analytic spread. They also cover algebroid curves and chain conditions, ideals and modules, and integral independence.

## An Introductory Course In Commutative Algebra

**Author :**A. W. Chatters

**ISBN :**0198501447

**Genre :**Mathematics

**File Size :**58. 98 MB

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This book is a concise and carefully written introduction to topics in commutative algebra, with an emphasis on worked examples and applications. The elegant algebraic theory is combined with applications to number theory, problems in classical Greek geometry, and the theory of finite fields, which has important uses in other branches of science. Topics covered include an introduction to rings and Euclidean rings, UFDs and PIDs, factorization of polynomials, fields and field extensions, and algebraic numbers. This book could form a springboard to further study of abstract algebra, but is also eminently suitable as the course text for an entire undergraduate course.

## Lectures On Rings And Modules

**Author :**Joachim Lambek

**ISBN :**082184900X

**Genre :**Mathematics

**File Size :**76. 35 MB

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Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. --Zentralblatt MATH 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.