# How to learn commutative algebra?

I want to learn commutative algebra from scratch. I was wondering, as you are experts in mathematics, what you think is the best way to learn commutative algebra? Is there any video course available for commutative algebra? Will there be some online course for commutative algebra on some website like Coursera, etc?

I know noncommutative algebra up to the Artin-Wedderburn Theorem. Also, I know group theory up to the Sylow theorems and Galois Theory. I also know some basic topology.

I'm new to this site so I don't know what tags I should add for this question. Please feel free to edit my question.

Edit 1: I want to learn commutative algebra for learning Algebraic Geometry.

• Why on earth do you want to learn commutative algebra ? May 29 '15 at 12:56
• I would recommend you to start with math.uiuc.edu/~r-ash/ComAlg.html May 30 '15 at 12:11

No doubt that Atiyah,Macdonald "Introduction to Commutative Algebra" is the classic on the subject. But if you are opting for self study, I would not recommend it. Usually commutative algebras are used in algebraic geometry but they are integral part of pure algebra too. But still the best way to learn is first do it in pure algebraic way and then as you will take topology, algebraic topology courses and other higher subjects towards algebraic geometry you will be comfortable with commutative algebra part.

So, My recommendation is you first take up "Undergraduate Commutative Algebra" by Miles Reid and skip Chapter $5$ if you do not want the flavour of Algebraic geometry right now, or can also go through it, as you like. Then the next step is Steps in commutative algebra by Sharp. After doing this second book, you will be good enough in commutative algebra to read whatever book/notes or research papers in the subject.

It is a really interesting subject. Make sure if you like this subject and want to stick with pure algebra instead of algebraic geometry or even both, Do read "A First Course in Noncommutative Rings " by T.Y.Lam if you get interested in Ring theory.

I would recommend first to work through Atiyah,Macdonald "Introduction to Commutative Algebra", ideally from cover to cover. Next get the three books:

1) Matsumura, "Commutative Algebra"

2) Serre, J.-P., "Algebre Locale Multiplicites"

3) Eisenbud, D. "Commutative Algebra with a view towards Algebraic Geometry"

and use them to deepen your understanding of the topics you have seen in Atiyah, Macdonald. In 2) already the introductory parts are a very worthwhile reading, the book is very good on graded and filtered rings, completions, dimension theory of noetherian local rings. The book 1) is useful for the theory of associated primes and primary decomposition of modules, and for a lot of more advanced topics up to excellent rings, formal smoothness, and also contains a presentation of kähler-differentials.

The book 3) is also a very good reading and full of topics and examples - I would consider it as indispensible. Helpful are also its appendices with an introduction of homological algebra, Ext and Tor and spectral sequences.

• Eisenbud is nice, but indeed not something to start with. I found myself going back to Dummit and Foote for some basic notions before moving further through Eisenbud. (this was for my bachelor thesis on Hilbert Series). The presentation is fast and touches upon details not too thoroughly for my taste, but complementing it with an advanced undergraduate book such as I did works out. On another note, Atiyah, Macdonald is great, but I did not have a copy of this book at the time. In hindsight, yes, definitely worth the purchase.
– user459879
Nov 24 '20 at 14:41

The book "Introduction to Commutative Algebra" (by Atiyah-Macdonald) is a good starting point, but if it seems difficult for you, you can consult the book "Steps in Commutative Algebra" (by Sharp), which goes into more detail.

Then you can continue with either
Bruns-Herzog's great book "Cohen-Macaulay rings" plus the book "Ideals, Varieties, and Algorithms" (by Cox D., Little J., O'Shea D.)
Or
David Eisenbud's great book "Commutative Algebra with a View Towards Algebraic Geometry".

Here, You can see Video lectures for Commutative Algebra; And here You can see Video lectures for Algebraic Geometry, see also here.

Here, you can see "Video lectures of mathematics courses available online for free"

• Good job on posting a link to some video resources. This is also what the OP requested. Much appreciated.
– user459879
Nov 24 '20 at 14:46
• You could also start with Ideals, Varieties, and Algorithms to be introduced to some commutative algebra alongside computational algebraic geometry. The formal prerequsites are linear algebra and some experience with proofs, so the OP looks well-prepared.
– J W
Jan 29 at 10:45

The best choice is "Algebraic Geometry and Commutative Algebra" by Siegfried Bosch. Textbooks of the German authors are typically very friendly to the user.

Modules are definitely the next topic to tackle on your way to understanding Commutative Algebra (after the topics you listed).

You may wish to check these commutative algebra notes by Pete L. Clark (university of Georgia) http://math.uga.edu/~pete/integral.pdf

• I am not sure what you mean by modules? Do you mean learning about modules? In that case it is not really a "way" to learn but a specific topic. May 29 '15 at 11:58
• Pete's notes are wonderful,crystal clear and completely modern-definitely worth checking out for the student serious about commutative algebra. I would worry a bit if they might be too advanced for the OP,though. May 29 '15 at 18:20
• I also recommend Dummit and Foote "Abstract Algebra" for the topic of modules. It is a bit dry, but I like their presentation (often very thorough and slow-paced). They also nicely tie in some category theory in relation to functors that preserve "exactness" of a short exact sequence of modules (in their presentation of projective, injective and flat modules).
– user459879
Nov 24 '20 at 14:36