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My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura (Commutative Ring Theory), and Eisenbud. There are also other books by Reid, Kemper, Sharp, etc. Can someone outline the differences between these texts, their relative strengths, and their intended audiences?

I am not listing my own background and strengths, on purpose, (a) so that the answers may be helpful to others, and (b) I might be wrong about myself, and I want to hear more general opinions than what might suite my narrow profile (e.g. If I said "I only like short books", then I might preclude useful answers about Eisenbud, etc.).

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    $\begingroup$ Eisenbud wrote his book as a preparation for Hartshorne, which is what "with a View Toward Algebraic Geometry" in the title alludes to. $\endgroup$ May 6, 2011 at 9:05
  • $\begingroup$ Are you recommending Eisenbud on those grounds, then? $\endgroup$ May 6, 2011 at 17:28
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    $\begingroup$ Yes, and also because I like Eisenbud's expository style: Also have a look at his little book "the geometry of schemes". $\endgroup$ May 6, 2011 at 18:55
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    $\begingroup$ Excellent question! $\endgroup$ Jun 5, 2011 at 11:32
  • $\begingroup$ Matsumura's book although can be a bit difficult for beginners, is actually quite good if you want to have as many knowledge in commutative algebra that was assumed in Hartshorne as possible before reading Hartshorne, and that many of the results in commutative algebra explicitly stated in Hartshorne's book also has a citation on Matsumura's book. On the other hand, I sometimes found the geometric explanations in Eisenbud's book a bit hard to grasp when I did not have any experience in any algebraic geometry; the book itself is also quite large (>550 pages). $\endgroup$ Aug 22, 2020 at 11:50

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I would recommend:

(1) Firstly, one should study field theory and Galois theory fairly thoroughly. The main reasons are:

a. Fields are the best understood examples of commutative rings from an ideal-theoretic point of view (a field has exactly two ideals) and field theory often motivates many important concepts in commutative algebra, e.g., modules (analogue in field theory: vector spaces) and integral extensions (analogue in field theory: algebraic extensions); also polynomial rings over fields are the best understood types of polynomial rings and are one of the main objects of study in algebraic geometry.

b. The applications of commutative algebra to algebraic number theory, for example, is very much based on Galois theory.

(2) Once one has a solid understanding of field theory and Galois theory, one can start learning commutative algebra. There are many good books on commutative algebra at the basic level. I recommend Atiyah and Macdonald's "An Introduction to Commutative Algebra" for the following reasons:

a. The book presents commutative algebra in a very elegant manner. I can assure you that if you read the entire book (~ 130 pages) and do all the exercises, you will have a very solid knowledge of commutative algebra.

b. The exercises are excellent and introduce the reader to many important concepts in commutative algebra not treated in the text, e.g. the spectrum of a ring, affine schemes, faithful flatness, direct limits, Hilbert's Nullstellensatz, Noether's normalization lemma etc. It is highly recommended that one does, or at least looks at, all of the exercises since approximately half of the material in the book is treated in the exercises. Many exercises have hints (which are almost always complete solutions) and thus the book is suitable for self-study.

Unfortunately, I have not read too many other introductory books on commutative algebra. "Algebra: A Graduate Course" by Martin Isaacs is also a good introduction to commutative algebra; however, the book is not one on commutative algebra purely. Similarly, Serge Lang's "Algebra" is also a good introduction.

(3) I think that there are three main choices for commutative algebra reading after Atiyah and Macdonald: "Commutative Algebra" by Hideyuki Matsumura, "Commutative Ring Theory" by Hideyuki Matsumura, and "Commutative Algebra: With a View Toward Algebraic Geometry" by David Eisenbud. Chapter IV of EGA is also a good reference if you are comfortable with the idea of reading French. However, I have not read any of these books (although I will do so soon) and thus I cannot comment further. Note that Matsumura's "Commutative Algebra" has very few exercises.

I hope this helps!

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    $\begingroup$ How could that not help? Thanks for the effort! $\endgroup$
    – t.b.
    Jun 5, 2011 at 12:08
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    $\begingroup$ @Andrew Dear Andrew, I'm really sorry I didn't catch your message; I was inactive around October last year! You've probably already figured out an answer to your question in the past nine months. However, if you're still interested in an answer to this question, then please let me know and I can try to give you a detailed recommendation. Indeed, if you have any questions about math textbooks in general, then I am happy to advise. Personally, I learnt Galois theory from "Algebra: A Graduate Course" by Isaacs but Dummit and Foote's "Abstract Algebra" is popular too! Regards, $\endgroup$ Jul 2, 2013 at 2:29
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    $\begingroup$ @AmiteshDatta Thanks for your warm and kind reply. Yes, for now I'm OK with texts. But I would like to take this opportunity to tell you how inspiring your endeavor is. I hope things are going well for you in all regards. Best, $\endgroup$
    – user12802
    Jul 2, 2013 at 10:40
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    $\begingroup$ @Andrew Dear Andrew, thank you so much for your kind words! I really appreciate them and I hope everything is going great for you too! Regards, $\endgroup$ Jul 2, 2013 at 15:26
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    $\begingroup$ +1 for Dummit, Foote. Though haven't worked it out myself but have heard that Field and Galois theory by Patrick Miranda is good too. $\endgroup$
    – Shreya
    May 25, 2018 at 17:00
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Eisenbud's book ("Commutative Algebra with a view towards algebraic geometry") is really gentle, and has good explanations together with lots of good illustrations. It is however huge, so you shouldn't really try to read the whole book.

Atiyah-MacDonald is no doubt the best mathematics book I've read, but it was a pain working through it because it is so densely written.

Edit: Some of the reasons I like Atiyah-MacDonald: * Tons of exercises - and really good ones too. * Short, concise language (this makes reading proofs a bit harder, but it pays off later when you're using the book as reference). * Very good choice of topics covered.

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    $\begingroup$ I wouldn't classify Eisenbud's book as gentle. $\endgroup$ Mar 24, 2021 at 2:31
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For a down-to-earth introduction, see A Singular Introduction to Commutative Algebra (also here). See also Reference book for commutative algebra from MO and a list but without comments.

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Since you've mentioned Sharp, let me write some thoughts about the book "Steps in commutative algebra".

This book, as Sharp says, is intended to be a preparatory book for other well-known books of Commutative Algebra like Matsumura and Atiyah-Mc Donald because its treatment is more elementary since the author doesn't discuss any topic of commutative algebra that requires homological ideas. So there are some topics not covered and others are studied only in a partial way, e.g., regular sequences and Cohen-Macaulay rings.

Although that, Sharp wrote a book which includes module theory as can be found in any book of basic abstract algebra, and I find the above practical because whenever I need a result relative to modules I can find it on his book.

In addition, the book by Sharp has many exercises among the different sections of each chapter which are helpful in order to understand well the given definitions and ideas. Moreover, Sharp has a very friendly way to develop the topics which helps to read and understand the theory.

Needless to say, I definitely recommend the book by Sharp because is suited for beginners in commutative algebra.

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Shameless plug: I wrote an introductory book on commutative algebra. It is available online and I hope it will not derail the discussion to add it as another alternative.

To give some context: I tried to make the book have really few prerequisites, and show the connection of commutative algebra with other fields, such as algebraic number theory, algebraic geometry and computational algebra. I avoided homological methods, because these will be treated in a subsequent volume, so this book does not contain anything about spectral sequences, flatness, Ext and Tor, regular sequences and the Koszul complex and so on.

But it contains the classical theory of Noetherian rings, including primary decomposition and dimension theory, and introduction to the study of Dedekind rings, especially the rings of integers in number fields, Groebner bases and resultants, the theory of valuations and absolute values, a detailed study of multiplicity, completions and more.

To get an idea of the level, it is a little more advanced (in parts) than Atiyah-MacDonald but it expands a lot more on connections with other parts of mathematics.

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It's a bit late. But since no one has mentioned it, I would mention Gathmann's lecture notes on Commutative Algebra (https://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013.pdf).

The exposition is excellent. The content is comparable to Atiyah-McDonald, but contains much more explanation. It emphasizes the geometric intuitions throughout the lectures. For example, the chapters on integral ring extension and Noetherian normalization have one of the best expositions of the geometric pictures behind these important algebraic concepts that I have read among several introductory books on commutative algebra. Chapters usually begin with a very good motivation and give many examples.

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