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基本説明
A textbook for advanced undergraduate students as well as graduate students and mathematicians seeking an accessible introduction to this fascinating area of abstract algebra.
Full Description
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
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: Grobner Bases 154 Solutions to Selected Problems 185 Suggestions for Further Reading 209 Index 213
