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I have been trying to relearn parts of algebra (mostly module theory and (advanced)linear algebra) from Lang, which, frankly, is not going too well.

Now, I have managed to get my hands on 'Aluffi - Algebra: Chapter 0'. And it (specifically the section called linear algebra reprise) seems pretty good. But, before I jump in and start studying from it, I want to make sure that I won't be repeating my experience with Lang. One indicator could be that it is a popular textbook. But, on searching, I found that it is not much used elsewhere.

My question:

Has anyone taken a formal/reading course using this book or studied it at length (not just leafed through it) and can thus recommend it and if it's possible (and not asking too much of their time) could they write briefly the course plan they followed?

Added This is in response to KCd's comments

What I am trying to do before next semester, is to take stock and make notes on some topics in Algebra generally taught in the first year of grad school (I am not in grad school. Will apply for Fall 2013). The notes will be (or on topics from) basic module theory, tensors, some exterior algebra, basic commutative algebra (an example of a topic would be Localization) (Some of these things I have learnt in courses such as Differential Geometry, etc.)

For instance, in the note which will end with a proof of the structure theorem over PIDs (hopefully one that I am truly comfortable with), I began with defining free modules via universal property, instead of how I had learned it from Artin's book.

Earlier, I thought I'll do this while relearning from Lang. But that's not serving me too well, for reasons I am not entirely clear about. Then I came upon Aluffi's book.

[@KCd: Let me also add that when I was trying to write up a note on basic Galois Theory, I came across your expository notes. They are spectacular.]

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I don't know Aluffi but Jacobson's Basic Algebra in two volumes is excellent and in a style totally different from Lang's. –  lhf Jun 22 '11 at 15:52
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If you already like the linear algebra part of Aluffi's book, just continue with modules and other linear algebra and see for yourself what the book is like. Whether it is a "popular" textbook or not isn't really so important; it's only 2 years old so it can't possibly yet appear popular compared with long-established texts. –  KCd Jun 22 '11 at 17:13
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Your question title is too broad compared with what it is you are trying to learn (algebra vs. modules and adv. linear algebra). Could you mention some topics that you are trying to relearn? –  KCd Jun 22 '11 at 17:15
    
Dactyl: since you mentioned the page where I posted a note on Galois theory, look there for notes on tensor products and exterior powers. –  KCd Jun 22 '11 at 19:59
    
If it isn't against site rules, would it be possible to link to the page where KCd's notes are on? I would be interested in these as well. –  james Jun 22 '11 at 21:40

2 Answers 2

I found Aluffi's book to be a refreshingly gentle advanced text on abstract algebra,Dactyl. It covers algebra in just the right amount of detail for a graduate course-not spoon-feeding, but not ultra-abstract or terse either-with a TON of good exercises. And it really tries to get students comfortable with using categories and commutative diagrams,too.

Unfortunately,my one beef with the book is that it's very classical. It doesn't cover a lot of topics that are very relevant today-such as commutative algebra beyond the bare bones, nothing on group representation theory (!), nothing on quantum groups,etc.But then,with the exception of Louis Rowen's awesome 2 volume course-which I strongly recommend as collateral reading-none of the standard graduate algebra texts do.

However,there IS something unique and cutting edge in the book that I don't think Aluffi gets a lot of credit for-there's a whole section on the elements of homotopic algebra, which is a very significant topic of current research interest and I don't think it's covered in any of the other standard graduate algebra books! But other then that-it covers the run of the mill stuff. It does it very well, but it's not very original in content with that one major exception. However,considering how well Aluffi presents the rest of the material, that one major exception makes the book worth having.

Overall, it's a VERY nice book indeed and supplementing it with Rowen's books or Atiyah/Macdonald and Reid and Paul Etingov's upcoming book on representation theory will give you everything you need for a terrific self-study course in algebra that covers all the essentials.

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Etingof and Reid are coauthoring a book on representation theory? Do you have a link? –  Akhil Mathew Jul 28 '11 at 5:35
    
@Akhil I think that "Reid" here refers to Miles Reid's Undergraduate Commutative Algebra text. And Etingof's book I believe is this one. Needless to say, there are four other authors that were omitted in the mention above. –  Adrián Barquero Jul 28 '11 at 6:25
    
@Akhil Actually this AMS page gives more information about the book. –  Adrián Barquero Jul 28 '11 at 6:30
    
@Adrian: Ah. Thanks. Note that the book is available at arxiv.org/abs/0901.0827; I wasn't aware it was going to be published. That's cool! –  Akhil Mathew Jul 28 '11 at 14:49
    
@Akhil This is merely a draft version-the version coming out next month from the AMS Student Library-which continues to produce OUTSTANDING textbooks!-is over double in length from the notes version. But the draft should give everyone a good idea whether or not the book is worth buying. I'd say from what I've seen,it's a definite yes. Not only is it beautifully and authoritatively written,it's a lot more up to date then the classical sources. –  Mathemagician1234 Aug 1 '11 at 4:11

Have a look at the review of it at http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=70277

Looks pretty nice, I may have to get it myself!

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A couple of comments: (i) I find the MAA reviews to be somewhat pollyannaish: only very rarely have I encountered an at all negative review. (ii) I have browsed through this book as well as its sequel and I am considering buying copies of my own. This testifies to its quality (I own several other standard "thick graduate algebra texts") and also to the fact that it covers some things that other books don't (especially the second volume). It strikes me as being one of the more graceful texts available but probably not the easiest. –  Pete L. Clark Jun 22 '11 at 19:23
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@Pete L. Clark: I am especially interested in your opinion. Would you expand on your comment. Thank you. –  Dactyl Jun 22 '11 at 19:55
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I would be interested in comments that compared the Aluffi text to Mac Lane-Birchoff's Algebra with respect to content and clarity; I have the latter and, from what I have read, I find it extremely lucid and well-written –  ItsNotObvious Jun 22 '11 at 21:08
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@Pete: What is the sequel? The theaters here are still showing The Great Train Robbery, so perhaps I am just behind the times. –  Jack Schmidt Jun 22 '11 at 21:36
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@Jack: Thanks for asking. After looking around for a bit, I discovered that I was confusing Aluffi's book with the two volume work by Falko Lorenz. I'm sure I must have flipped through Aluffi's book at some point, but nothing much comes to mind. (Oops.) –  Pete L. Clark Jun 22 '11 at 21:44

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