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.]
Best Answer
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.