What do you think?
Rate this book
Introduction To Commutative Algebra
This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization.
140 pages, Hardcover
First published January 1, 1969
21 people are currently reading
241 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 10 of 10 reviews
July 26, 2020
This shit made me cry, but at least It got me some decent grades.
April 8, 2016
Read only the first couple of chapters. Proves strong results, with concise proofs. Not much intuition. Good exercises.
December 24, 2021
Absolutely necessary trial by fire.
September 11, 2021
As the description suggests, this book serves as a rapid introduction to some of the main topics in commutative ring theory but is not a replacement (this is important!) for the more voluminous texts such as Zariski-Samuel, Bourbaki, Matsumura, or Eisenbud. However, I think this book is a must read for anyone's first introduction to the subject; it quickly develops much of the basic theoretical framework and language needed to explore topics in algebraic geometry and number theory.
This book is extremely well-written. There are very little errors, and most of them don't affect the logical flow of the text or create any critical errors in the proofs provided (an errata for the book can be found on MathOverflow). Each proof is short and succinct while remaining fully comprehensible, which no huge gaps in logic that the reader is expected to fill. Examples are carefully chosen, and conceptual transitions are smooth. There is also a wealth of exercises at the end of each chapter which supplement the main material beautifully.
Completing at least some of the exercises to this book are essential; around 30-40% of the theory is treated in the exercises. Furthermore, the exercises are essential for building one's intuition and big-picture view of the topics presented, since much of the algebro-geometric and number-theoretic connections are presented there. Hilbert's Nullstellenstatz and Noether's Normalization Lemma, for example, are treated in the exercises of Chapters 5 and 7. I would recommend first-time readers to (at the very least) complete every exercise in Chapter 1. In essence, this is not a book you can just read; it is a book you have to work through.
However, there are some things one must know about this book before diving in. This book is extremely dense (though not the worst I've seen). A ton of content is crammed into just 126 pages, and since there are very little computational examples, novel material can become difficult to digest. I had to drop this book the first time I tried to tackle it; I managed to get through 75% of Chapter 3 before I realized that I lacked intuition for the brand-new, unfamiliar concepts which were being used repeatedly.
Most of the conceptual hurdles happened in chapter 1, 2, and 3. The groundwork for all subsequent theory in the book is laid here, and thus (in my opinion) it is paramount for any reader to fully master the concepts presented here and to integrate them with their standard mathematical toolkit. The concepts that were novel to me and responsible for most of my early headaches were tensor products, algebras, and localization (or just forming rings of fractions in general). It also took a while before many of the ideal operations introduced in Chapter 1 became instinctive. The key theme is time; these concepts were new to me, and they weren't things I could pick up in a day or two.
Another downside of this book is the lack of motivation for many of the concepts presented. For anyone who isn't learning this text alongside anything else, do yourself a favor and skim the first few chapters of any classical algebraic geometry book (e.g. Beltrametti, Shafarevich) or the first few sections of Hartshorne; even just a surface-level understanding of how commutative algebra is used in algebraic geometry will illuminate many of the ideas in the book, and the authors' brief references to algebro-geometric or number-theoretic concepts, which they use as motivation, will make much more sense.
If one can make it past the initial hurdles, however, the book will become much more enjoyable. Though it felt fast-paced at the beginning, I quickly began to admire the presentation style. The proofs are beautifully written, with no wasted words and not overly omissive. Almost every proof is only a few lines long. I have never before seen a textbook that has managed to accomplish all these things simultaneously (although my experience is admittedly limited), and I'm not confident I'll ever see one again. Amazing. 5/5
This book is extremely well-written. There are very little errors, and most of them don't affect the logical flow of the text or create any critical errors in the proofs provided (an errata for the book can be found on MathOverflow). Each proof is short and succinct while remaining fully comprehensible, which no huge gaps in logic that the reader is expected to fill. Examples are carefully chosen, and conceptual transitions are smooth. There is also a wealth of exercises at the end of each chapter which supplement the main material beautifully.
Completing at least some of the exercises to this book are essential; around 30-40% of the theory is treated in the exercises. Furthermore, the exercises are essential for building one's intuition and big-picture view of the topics presented, since much of the algebro-geometric and number-theoretic connections are presented there. Hilbert's Nullstellenstatz and Noether's Normalization Lemma, for example, are treated in the exercises of Chapters 5 and 7. I would recommend first-time readers to (at the very least) complete every exercise in Chapter 1. In essence, this is not a book you can just read; it is a book you have to work through.
However, there are some things one must know about this book before diving in. This book is extremely dense (though not the worst I've seen). A ton of content is crammed into just 126 pages, and since there are very little computational examples, novel material can become difficult to digest. I had to drop this book the first time I tried to tackle it; I managed to get through 75% of Chapter 3 before I realized that I lacked intuition for the brand-new, unfamiliar concepts which were being used repeatedly.
Most of the conceptual hurdles happened in chapter 1, 2, and 3. The groundwork for all subsequent theory in the book is laid here, and thus (in my opinion) it is paramount for any reader to fully master the concepts presented here and to integrate them with their standard mathematical toolkit. The concepts that were novel to me and responsible for most of my early headaches were tensor products, algebras, and localization (or just forming rings of fractions in general). It also took a while before many of the ideal operations introduced in Chapter 1 became instinctive. The key theme is time; these concepts were new to me, and they weren't things I could pick up in a day or two.
Another downside of this book is the lack of motivation for many of the concepts presented. For anyone who isn't learning this text alongside anything else, do yourself a favor and skim the first few chapters of any classical algebraic geometry book (e.g. Beltrametti, Shafarevich) or the first few sections of Hartshorne; even just a surface-level understanding of how commutative algebra is used in algebraic geometry will illuminate many of the ideas in the book, and the authors' brief references to algebro-geometric or number-theoretic concepts, which they use as motivation, will make much more sense.
If one can make it past the initial hurdles, however, the book will become much more enjoyable. Though it felt fast-paced at the beginning, I quickly began to admire the presentation style. The proofs are beautifully written, with no wasted words and not overly omissive. Almost every proof is only a few lines long. I have never before seen a textbook that has managed to accomplish all these things simultaneously (although my experience is admittedly limited), and I'm not confident I'll ever see one again. Amazing. 5/5
December 11, 2020
in the abstract language of nonsense - this is a good book.
October 1, 2021
A slim book of barely more than 100 pages—a brain-cracking masterwork of brevity/density that goes from prime ideals to tensor products and flat modules in less than 10 pages. Ferocious. A classic.
June 1, 2022
a good complementary book for first graduate algebra course
March 2, 2017
This book is by Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng. I really wish I had studied this more carefully and appreciated this subject.
January 27, 2018
Evergreen classic.
Currently reading
May 10, 2018I want to read this book. Anyone interested in joining me to study together, let me know. We can also look for expository work afterwards. Just message me.
Displaying 1 - 10 of 10 reviews