Topics in Commutative Ring Theory

By John J. Watkins

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.

• Language: English
• Publisher: Princeton University Press
• Release date: Feb 9, 2009
• ISBN: 9781400828173

John J. Watkins

John J. Watkins is professor emeritus of mathematics at Colorado College. His books include Across the Board: The Mathematics of Chessboard Problems (Princeton), Topics in Commutative Ring Theory (Princeton), Graphs: An Introductory Approach, and Combinatorics: Ancient and Modern.

Book preview

TOPICS IN COMMUTATIVE RING THEORY

TOPICS IN

COMMUTATIVE

RING THEORY

JOHN J . WATKINS

PRINCETON UNIVERSITY PRESS

Princeton and Oxford

© 2007 by Princeton University Press

Published by Princeton University Press, 41 William Street,

Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press,

3 Market Place, Woodstock, Oxfordshire OX20 1SY

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Watkins, John J.

Topics in commutative ring theory / John J. Watkins.

p. cm.

Includes bibliographical references and index.

eISBN: 978-1-40082-817-3

1. Commutative rings. 2. Rings (Algebra) I. Title.

QA251.3W38 2007

512.44–dc222006052875

British Library Cataloging-in-Publication Data is available

This book has been composed in ITC Stone Serif

Printed on acid-free paper. ∞

pup.princeton.edu

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

For Jim Brewer

Contents

Preface

1 Rings and Subrings

2 Ideals and Quotient Rings

3 Prime Ideals and Maximal Ideals

4 Zorn’s Lemma and Maximal Ideals

5 Units and Nilpotent Elements

6 Localization

7 Rings of Continuous Functions

8 Homomorphisms and Isomorphisms

9 Unique Factorization

10 Euclidean Domains and Principal Ideal Domains

11 Polynomial Rings

12 Power Series Rings

13 Noetherian Rings

14 Dimension

15 Gröbner Bases

Solutions to Selected Problems

Suggestions for Further Reading

Preface

This collection of lectures in commutative ring theory has grown out of a course I have taught for many years at Colorado College for advanced undergraduates taking a second course in abstract algebra and is intended as an introduction to abstract mathematics. It is abstraction —more than anything else—that characterizes the mathematics of the twentieth century. There is both power and elegance in the axiomatic method, attributes that can and should be appreciated by students early in theirmathematical careers and even if they happen to be confronting contemporary abstract mathematics in a serious way for the very first time.

Commutative ring theory arose more than a century ago to treat age-old questions in geometry and number theory; it is is therefore, in part, a branch of applied mathematics in the sense that it is applied to other areas of mathematics. Even today it draws nourishment from these two subjects. But commutative ring theory is also very much a part of pure mathematics, and as such it has a life of its own that is quite independent of its origins. It is largely the balance — even the tension —between these two aspects of its personality thatmakes commutative ring theory such a rich and beautiful subject for study.

While many readers of this book may well have previously studied modern algebra, I will assume no particular knowledge on the part of the reader other than perhaps a passing awareness of what a group is. I believe that, by focusing our attention on a single relatively narrow field of modern abstract mathematics, we can begin at the beginning and take an enthusiastic reader on a trip far into the vibrant world of contemporary mathematics. The itinerary and pace for this journey have been conceived with advanced — and, I repeat, enthusiastic —undergraduates in mind, but I sincerely hope that graduate students beginning their specialization in algebra as well as seasoned mathematicians from other areas of mathematics will also find the journey worthwhile and pleasurable.

The intent, then, is a fairly leisurely and reader-friendly passage. Our goal is to get a feel for the lay of the land, tomarvel at some of the vistas, and to poke around a few of the back roads. We may miss a couple of the main highways, and we will certainly resist climbing some of the higher peaks.We may not get as far, or as high, as some would like, but we prefer not to lose anyone along the way. There will always be time —and other guidebooks—for other,more ambitious, trips for those of you who have enjoyed this one.

I have placed a series of problems at the end of each chapter in order to encourage active reading. These exercises provide you with a way of immediately reinforcing new concepts, as well as becoming adept at some of the fundamental techniques of commutative ring theory. I have provided solutions to some of these exercises at the end of the book so that you can compare your work with what could be considered standard solutions. Some of the problems are routine exercises designed to build technical skill or reinforce basic new ideas, and advanced readers may well wish to skip most of them. However, many of the problems are in fact extremely important mathematical results in their own right, and will be used freely later on in the text.

I would like to thank first of all the many students at Colorado College who have been subjected to earlier, considerably rougher, versions of this book. In particular, I appreciate the enthusiastic and generous help I have received through the years from David Carlson, T.J. Calvert, Lisa Converse, Courtney Gibbons, Laura Hegerle, Eric Raarup, Karin Reisbeck, Chantelle Szczech-Jones, Mark Sweet, Rahbar Virk, and Trevor Wilson. I would also like to thank two colleagues, Doug Costa and Wojciech Kosek, whose suggestions have improved this book considerably. Mostly, though, I would like to thank Jim Brewer,my thesis advisor at the University of Kansas, from whom I not only first learned this beautiful subject but also received so very much in terms of guidance, wisdom, and friendship.

John J.Watkins

Colorado Springs

May 31, 2006

TOPICS IN

COMMUTATIVE

RING THEORY

1

Rings and Subrings

The Notion of a Ring

In 1888 — when he was only 26 years old — David Hilbert stunned the mathematical world by solving the main outstanding problem in whatwas then called invariant theory. The question that Hilbert settled had become known as Gordan’s Problem, for it was Paul Gordan who, 20 years earlier, had shown that binary forms have a finite basis. Gordan’s proofwas long and laboriously computational; there seemed little hope of extending it to ternary forms, and even less of going beyond. We will not take the time here to explore