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John J. Watkins, "Topics in Commutative Ring Theory"

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

Summary: Wonderful!

Rating: 5

Rudin's "Principles of Mathematical Analysis" is as to an introduction of real analysis as this book is as to an introduction to Ring Theory. I personally call it a "Baby Ring Theory", and it was a wonderful mother in my small upbringing in algebra. I loved it and still do like having it on my bookshelf.

Summary: The best introduction to the subject I've ever seen.

Rating: 5

This is a tyny introductory book on commutative rings with a somewhat narrow coverage: while Krull dimension and Groebner basis are explained, Dedekind domain is not covered. Module theory is not treated, hence no homological algebra is touched.

But the treatment is by no means shallow, and practially every detail is filled in.

I highly recommend this book to those who have been fed up with the Bourbaki-style writing in which things are written in a "as short as possible" fashion.

Summary: Well-written and a lot of fun (for a math book!)

Rating: 5

I wanted to wait until I finished the book, but since there are no reviews... This is a treasure of a book. The book itself is very well made, well-bound, and uses good quality paper. The print is very clear and easy to read, (no clutter). The basic format is a few pages explaining a topic on Rings, some proofs and examples as you go along, and then problems for each chapter. The explanations are short and sweet, and the material flows from one chapter to the next. The symbology is explained and is consistent. I have since picked up several Abstract Algebra books and they all suffer from too much detail.

The fun comes from working on the problems, which are used to advance the topics. This book is a great mix of just enough background info, some discovery style learning, (with a lot of the problems worked out in the back when you get stuck). I started this book without any Abstract Algebra background, and am now over half-way through. My proof writing skills have improved dramatically as well. The author clearly spent a lot of time putting together this treasure of a book. So buy it now.

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Rating:
2.5 out of 5 by Book123