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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.

- $M$ finitely generated if submodule and quotient are finitely generated.
- Infinite Degree Algebraic Field Extensions
- let $H\subset G$ with $|G:H|=n$ then $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ (Dummit Fooote 4.2.8)
- Proper subgroups of non-cyclic p-group cannot be all cyclic?
- $F:G\to B$ is an isomorphism between groups implies $F^{-1}$ is an isomorphism
- Find the polynomials which satisfy the condition $f(x)\mid f(x^2)$

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

- If $I = \langle 2\rangle$, why is $I$ not a maximal ideal of $\mathbb Z$, even though $I$ is a maximal ideal of $\mathbb Z$?
- free subgroups of $SL(2,\mathbb{R})$
- Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
- Intuition, proof, one-sided group definition - Any set with Associativity, Left Identity, Left Inverse is a Group - Fraleigh p.49 4.38
- Unique normal subgroups of every possible order
- Isomorphism between $\mathbb R^2$ and $\mathbb R$
- Is Pythagoras the only relation to hold between $\cos$ and $\sin$?
- Direct product and normal subgroup
- what is the tensor product $\mathbb{H\otimes_{R}H}$
- Subgroups of $S_4$ isomorpic to $S_2$

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.

I hope this will help you!

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.

Modules are definitley the way to go about learning Commutative Algebra. check http://math.uga.edu/~pete/integral.pdf

The book “*Introduction to Commutative Algebra*” (by *Atiyah-Macdonald*) is a good starting point; but if it seems difficult for you, you can consult with the book “*Steps in Commutative Algebra*” (by *Sharp*), which goes more in details.

Then you can continue with either

*Bruns-Herzog*‘s grate book “*Cohen-Macaulay rings*” **plus** the book “*Ideals, Varieties, and Algorithms*” (by *Cox D., Little J., O’Shea D.*)

Or

*David Eisenbud*‘s grate book “**Commutative Algebra** *with a View Towards Algebraic Geometry*“.

See also here.

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”

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

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