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Topics in Commutative Ring Theory
John J. Watkins

Hardcover | 2007 | $99.95 | £83.95 | ISBN: 9780691127484
232 pp. | 7 x 10
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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


"As an honest, focused treatment of an important subject packaged into an attractive, slender volume that average undergraduates may reasonably hope to master in one semester, this book will find its niche and its admirers."--D.V. Feldman, Choice

"A highlight is the attention given here to the ring of continuous functions, an important non-standard example of a commutative ring that shows the profound relationship between algebra and topology."--Marco Fontana, Mathematical Reviews


"A very elementary introduction to commutative ring theory, suitable for undergraduates with little background. It is written with great care, in a conversational and engaging style that I think will appeal to students. Essentially every detail is made explicit, and readers are admonished to beware typical pitfalls. The book is also peppered with very nice detours into the history of mathematics."--Karen Smith, University of Michigan

Table of Contents:

Preface ix
CHAPTER 1: Rings and Subrings 1
CHAPTER 2: Ideals and Quotient Rings 11
CHAPTER 3: Prime Ideals and Maximal Ideals 23
CHAPTER 4: Zorn’s Lemma and Maximal Ideals 35
CHAPTER 5: Units and Nilpotent Elements 45
CHAPTER 6: Localization 51
CHAPTER 7: Rings of Continuous Functions 69
CHAPTER 8: Homomorphisms and Isomorphisms 80
CHAPTER 9: Unique Factorization 89
CHAPTER 10: Euclidean Domains and Principal Ideal Domains 100
CHAPTER 11: Polynomial Rings 110
CHAPTER 12: Power Series Rings 119
CHAPTER 13: Noetherian Rings 128
CHAPTER 14: Dimension 137
CHAPTER 15: Gröbner Bases 154
Solutions to Selected Problems 185
Suggestions for Further Reading 209
Index 213

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