Topics in Commutative Ring Theory

by Watkins, John J.

• ISBN13: 9781400828173
• ISBN10: 1400828171
• Copyright: 2008-09-02
• Publisher: Princeton Univ Pr

Summary

Topics in Commutative Ring Theoryis 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 Theoryis an ideal resource for anyone seeking entry into this stimulating field of study

Table of Contents

Preface	p. ix
Rings and Subrings	p. 1
Ideals and Quotient Rings	p. 11
Prime Ideals and Maximal Ideals	p. 23
Zorn's Lemma and Maximal Ideals	p. 35
Units and Nilpotent Elements	p. 45
Localization	p. 51
Rings of Continuous Functions	p. 69
Homomorphisms and Isomorphisms	p. 80
Unique Factorization	p. 89
Euclidean Domains and Principal Ideal Domains	p. 100
Polynomial Rings	p. 110
Power Series Rings	p. 119
Noetherian Rings	p. 128
Dimension	p. 137
Gröbner Bases	p. 154
Solutions to Selected Problems	p. 185
Suggestions for Further Reading	p. 209
Index	p. 213
Table of Contents provided by Publisher. All Rights Reserved.