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I am currently doing a one semester course on groups and rings where we have learned about (so far):

Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, isomorphisms, The Correspondence Theorem, Product and Quotient Groups. As of yesterday's lecture we learned about the First Isomorphism Theorem and a little bit about rings.

By the end of the course we should have done rings, endomorphisms, The Orbit-Stabilizer Theorem and subjects which I am not sure about.

I am wondering if this would be sufficient to start Atiyah Macdonald; I have opened the first few pages and it looks hard. For those who have done it, what do you think are the prerequisites before doing this? Perhaps something like Herstein's Topics in Algebra?

Thanks.

2 Answers 2

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Algebra Prerequisites: A knowledge of the following results:

(1) The definition of ring, subring, ideal and quotient ring.

(2) The correspondence theorem in ring theory.

(3) The notion of a prime ideal, of a maximal ideal, and the facts that an ideal $I$ of a commutative ring $A$ is prime (resp. maximal) if and only if $R/I$ is an integral domain (resp. field).

In short, the first 4 pages of Atiyah and Macdonald should be in the nature of a review for you.

(4) An extensive knowledge of field theory and Galois theory; for example, in addition to the elements of Galois theory, you should probably be familiar with separable and inseparable extensions and transcendental extensions. (Chapter 5 on integral extensions of commutative rings is better appreciated if you have already studied the theory of algebraic extensions of fields. Transcendental extensions are discussed in the chapter on dimension theory. Finally, at least one result in chapter 9 (on Dedekind domains) and a few exercises in chapter 5 require a knowledge of separable and inseparable extensions in field theory.)

(5) I think the Jordan-Hölder theorem in group theory is alluded to at some point in the text. (The discussion of modules of finite length in chapter 6.)

Topology Prerequisites: A knowledge of the following results:

(1) The definition of a topological space, of open and closed sets, of a basis for a topology, of compact sets and Hausdorff spaces, of subspaces, and of continuous functions. (In the text itself, point-set topology is most prominent in the chapter on completions but you will need point-set topology for the exercises as well.)

(2) Urysohn's lemma is needed to solve exercise 4 in chapter 4 and at least one exercise in chapter 1 (on the characterization of the maximal spectrum of a commutative ring).

Summary: The most important prerequisites are point-set topology and the theory of fields. You can read chapters 1-4 of Atiyah and Macdonald with only (1)-(3) of the Algebra Prerequisites above and chapter 5 of Atiyah and Macdonald with a knowledge of algebraic and separable extensions of fields. Chapter 9 of Atiyah and Macdonald also requires a knowledge of separable extensions of fields and chapter 10 of Atiyah and Macdonald requires a knowledge of (1) of the Topology Prerequisites above.

However, in order to do the exercises in Atiyah and Macdonald (which are the most important part of the text, in my opinion), you will need all the prerequisites above. Point-set topology is an essential prerequisite in the exercises because many exercises discuss affine schemes. Also, the elements of Galois theory are needed in some exercises in chapter 5, for example.

I hope this helps!

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    Thanks, by the way I think this would mean reading a lot on galois theory and things like that, which would mean at least Algebra II for me.2011-08-30
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    As for the topology prerequisites maybe Munkres can help to cover that...2011-08-30
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    @DBLim I think chapters 2 and 3 of Munkres' textbook should furnish more than enough preparation in regard to point-set topology. However, you probably do not need to wait until Algebra II to learn Galois theory (you could learn Galois theory in the summer).2011-08-31
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    +1 for mentioning the exercises. The amount of commutative algebra one learns from this small, slender, book, with its hundreds of exercises, has always fascinated me. Easily one of the best math books I've ever read.2011-08-31
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    @Amitesh Datta Rudin's chapter 2 has some basic rudiments of topology...2011-08-31
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    @DBLim Indeed, but you will need to learn about general topological spaces as well as about metric spaces. The Zariski topology is quite a "strange topology" as it is (in most cases of interest) non-Hausdorff. In fact, if $A$ is a commutative ring, then the Zariski topology on $\text{Spec}(A)$ is Hausdorff if and only if $A/\sqrt{0}$ is absolutely flat.2011-08-31
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    @DBLim OK, we can continue this discussion by email if you wish; please let me know when you have read my comment directly above and I will delete it.2011-08-31
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    @Amitesh Datta I have read your comment, I may send you an email soon. Ben2011-08-31
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    As your list is pretty detailed and nearly complete, let me mention a missing detail: basic knowledge of homological algebra, especially the Tor (and Ext if I remember this correctly) functor(s) are needed for some (few) exercises around exactness. Anyway, @BenjaLim, there is something you need if you don't want working through it become hard — experience. It is a highly recommended book for self-study! But rather for more experienced students who want to become well prepared for algebraic geometry than for someone who recently took his first steps in algebra.2013-01-23
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    Hi @Ben, thank you so much for your comment! I didn't notice it earlier, and I apologise for my slow reply. You are absolutely right that a knowledge of the Tor and Ext functors is required for some of the exercises, especially on flatness. I didn't know any homological algebra when I first read the textbook, so I just ignored those exercises, but I now know that it is not that hard to pick up what is required (and it is well worth it in the long run). I recall Matsumura's *Commutative Ring Theory* furnishes a quick introduction to the required homological algebra in one of the appendices.2014-07-13
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    Dear @Amitesh, you're welcome! When I wrote the comment, I just wanted to add this little detail to your excellent answer. Anyway, you're right, I guess I should've added that it's no big deal to skip the few exercises where Tor and Ext are needed at first, because homological algebra is probably not meant to be done before one is comfortable with commutative algebra.2014-07-13
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You don't actually need a lot of abstract algebra knowledge before reading A-M. You will need to know the definitions of ideals, fields, and some basic group theory.

A-M is a hard book, and reading it is a pain, and so I cannot really recommend it for self-study. It is however a good book, one of the best I've read.

You could, for example, read it alongside Eisenbuds "Commutative Algebra with a view towards algebraic geometry", which is a really good (though enormous) book.

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    That's what I've heard a lot of people say that A-M is very dry, what about this Eisenbud text?2011-08-30
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    @Lim: I wouldn't call A-M dry. It is "dense". Eisenbud is very vividly written with more examples than A-M and good exercises (his presentation is also more geometric).2011-08-30
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    My god Eisenbud's is about 600 pages long!2011-08-30
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    @DBLim The prerequisites for reading Eisenbud's textbook are mostly subsumed in the appendices at the back of the book. However, you will need to spend at least a reasonable amount of time reading the appendices. For example, you will need to learn about free modules, projective modules, injective modules, complexes, the Tor and Ext functors and some multilinear algebra. The book also emphasizes computation (and has a chapter devoted to Grobner bases) but this is certainly not a bad thing (computational commutative algebra is very interesting).2011-08-31
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    @DBLim However, I would personally recommend you to read Atiyah and Macdonald's *An Introduction to Commutative Algebra*; in approximately 130 pages, you will learn much more than you will in 130 pages of Eisenbud's textbook. If you wish, you can read Eisenbud later (Eisenbud has explained in the beginning of his book how someone who has already read Atiyah and Macdonald's textbook can read his textbook). On the other hand, Matsumura's *Commutative Algebra* is also an excellent textbook (which I would recommend); a knowledge of its contents is essential for some parts of algebraic geometry.2011-08-31
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    I find Matsumura dryer than Atiyah-Macdonald though..2011-08-31
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    @Amitesh: I don't think a page of Eisenbud and a page of Atiyah–Macdonald are comparable in any meaningful sense. It takes me longer to read a page of AM because it's so dense! And a page full of exercises takes much much much longer to ‘read’ than a page of proof because one has to fill in the details...2011-08-31
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    @Zhen I do not think that the words "dry" and "dense" have any meaningful sense! I thought Atiyah and Macdonald was quite a readable book with plenty of explanations; it is true that reading a page of exercises takes time. However, I am not willing to enter a discussion about this kind of thing; I stated my opinion about a few textbooks and you are free to disagree with that opinion. In any case, one should not be "reading" Atiyah and Macdonald; one should be proving all the theorems of Atiyah and Macdonald on one's own. The proofs and results in the textbook are fairly straightforward.2011-08-31
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    @trony I generally do not like the word "dry" in mathematics. The elegance, beauty and interest of a subject should be self-evident from the exposition; if an author needs to "motivate" a subject, then the subject itself is not as interesting to you as it should be. Commutative algebra is a beautiful subject and one can see that on one's own by reading Matsumura; the elegance and beauty of the results and proofs in Matsumura are self-explanatory.2011-08-31
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    @Amitesh: Well, I was merely offering an opposing opinion, from the point of view of an undergraduate mathematician with a more conventional background. Not everyone has had the benefit of learning so much, whether by their own efforts or otherwise, by the age of 16!2011-08-31
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    @Zhen Dear Zhen, this is fine, of course! I understand what you are saying and I certainly agree that it is not meaningful to compare pages of two different textbooks. However, at the same time, there is always a degree of imprecision when one gives opinions about textbooks (for example, I have never seen the words "dry" and "dense" defined ...). I think the point is that Eisenbud is somewhat longer not only because he gives more explanations than Atiyah and Macdonald but also because he digresses frequently. In some sense, Atiyah and Macdonald is more to the point and this is what I meant.2011-08-31
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    @Zhen I hasten to add, however, that many comments that I have made about Eisenbud are rather unqualified as I have not read the entire textbook; needless to say, Eisenbud's textbook is wonderful and an outstanding addition to the literature. However, I personally found Atiyah and Macdonald's textbook more readable for the reason that the prose is rather succinct. I also prefer Matsumura's *Commutative Algebra* for the same reason.2011-08-31