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(Cross-posted at Math.Stackexchange)

I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. I've thought of the aforementioned texts, but additional suggestions would be welcome too.

Some points about my background: I’ve taken a course in Linear Algebra where I read Roman’s Advanced Linear Algebra in addition to Halmos’ Finite Dimensional Vector Spaces and Axler’s Linear Algebra Done Right as primary texts. I’ve already had a course of Abstract Algebra from Artin’s Algebra. Additionally, I have finished the first 7 chapters of Baby Rudin, and plan to try Loomis and Sternberg’s Advanced Calculus next.

These upcoming 3 semesters I have courses in Algebra (the syllabus for which is attached at the end). The prescribed texts are Jacobson’s Basic Algebra I, II, and Lang’s Algebra. Some of the material is familiar, so I’m looking to self-study beyond the syllabus.

I've looked through Lang's Algebra, MacLane and Birkhoff’s Algebra, and Jacobson’s Basic Algebra I,II. So far, Basic Algebra I seems much easier and more ‘leisurely’ than the other two. Understanding the exposition was not an issue for any of the books (I used G. Bergman's Companion to Lang for some assistance with Lang's Algebra). Unfortunately Basic Algebra I usually gives explicit constructions as opposed to using categories or universal properties. I would prefer to learn Abstract Algebra using Category Theory and Universal Properties openly; to do this from Jacobson’s book, I would have to use both volumes together. I’m not sure how to do this.

I referred to the Chicago Undergraduate Mathematics Bibliography, which suggested that a few portions from Basic Algebra II (such as Group Representation Theory) were best done from elsewhere.

Could somebody please compare using Basic Algebra I, II , MacLane and Birkhoff’s Algebra , and Lang’s Algebra. In particular, please explain their relative merits/demerits, levels of difficulty, ‘modernness’ of the treatment, and quality of the exercises? I enjoy struggling through the texts that are terse, and leave significant gaps (such as in proofs) for the reader to fill in, something like Rudin or Lang's books.

Lastly, is it better to do any one of these books from cover to cover? Or is it better to do individual sections from each book, or perhaps one book followed by another? If it is the latter two, then could the relevant chapters/order please be pointed out to me?

All comments and answers are greatly appreciated. Thank you so much for your time!

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Syllabus

  • Rings, ideals, homomorphisms, quotient rings, fraction fields, maximal ideals, factorization, UFD, PID, Gauss Lemma, fields, field extensions, finite fields, function fields, algebraically closed fields.
  • Galois theory: separable and normal extensions, purely inseparable extensions, fundamental theorem of Galois theory.
  • Module theory, structure theorem for modules over PIDs.
  • Multilinear algebra: tensor, symmetric and exterior products, tensor product of algebras.
  • Categories and functors, some notions of homological algebra.
  • Non-commutative rings, semisimplicity, Jacobson theory, Artin-Wedderburn theorem, group-rings, matrix groups, introduction to representations.
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  • $\begingroup$ Ps. I had asked this question on MathOverflow too, where it was suggested that I try posting this here on MathEducators StackExchange. If there are any issues, please advise me on how to modify/delete this question. Many thanks for your consideration! $\endgroup$ – Ishan Jun 18 at 13:21
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    $\begingroup$ Dear @TommiBrander Thanks for you comment. Unfortunately in the 24 days since it was posted, the question on MSE hasn't yet received an answer. $\endgroup$ – Ishan Jun 18 at 17:48
  • $\begingroup$ @Tommi, waiting 24 days to cross post here (more than three weeks) satisfies your recommendation, since you seem to have written your comment without knowing the OP satisfied the recommendation you give. So I hope you see the inapplicability of your comment here in this case. $\endgroup$ – Namaste Jun 18 at 20:59
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    $\begingroup$ Oh, sorry about that. My bad. $\endgroup$ – Tommi Jun 19 at 6:07
  • $\begingroup$ I haven't seen any mention of Aluffi's "Algebra: Chapter 0" yet so I'll throw it out here. As a secondary comment, my graduate coursework used an unpublished text which covers all of that material starting with category theory, but I'm not at liberty to share it. $\endgroup$ – Opal E Jun 19 at 15:30
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Search [mathematics.se] using the (reference-request) tag, and the (abstract-algebra) tag. There have been a number of posts, from students at various levels of study in abstract algebra, seeking text recommendations. One such post, mentions Lang's Algebra. It's a very thorough text (scan the Table of Contents here, but doesn't cover category theory explicitly.)

Another [mathematics.se] post of possible interest to you is Is Serge Lang's Algebra still worth reading?. Jacobson's texts are both also recommended, as is Lang's text, in that post, as are other texts you haven't mentioned.

In addition, see a question on [mathematics.se] which addresses MacLane and Birkoffs' Algebra and it's suitability as a text for upper-level undergrads and/or graduate students. (There are additional posts on [mathematics.se] about this text. This text seems to include more of the items listed in your syllabus, than the other options.

I think all three choices you list are fine choices. I'd suggest browsing through each, to discern which style, approach(es) best suit(s) your needs. My decisions, when it comes to self-study, inevitably involves the use of two highly recommended texts. Inevitably, one of them emerges as by "basis text", but when struggling with a particular topic, the second text is very useful to have. I just would not recommend juggling too many texts at one time.

Because of the level at which you want to self-study Algebra, I came across a rather highly regarded but classic text in pdf format: Advanced Modern Algebra: Part 1 by Joseph Rotman. (There's an associated text "Part 0" in the series, to be read first, which is Martin's Algebra text. There's also the sequel (3rd edition) Advanced Modern Algebra: Part 2. Combining Parts I and II, the topics covered are shown in the TOC. In a library, you may only find Rotman's 2nd Edition Advanced Modern Algebra, which covers much of Part I and II, in one volume.

All I see missing from these texts is an explicit chapter/coverage on Non-commutative Rings. If you can master the other topics, and have time to "taste"/sample non-commutative Algebra: Review this, this, or this.

I've used all of the texts I discuss here, (and Artin's Notes on non-commutative algebra), and the choice sometimes comes down to a text that students can use over two semesters studying Algebra, and/or the level of the course and the caliber at which students are at, etc.

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    $\begingroup$ If the first sentence missing something? $\endgroup$ – Tommi Jun 18 at 16:55
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    $\begingroup$ Dear @Namaste Thanks so much for your answer. It's been 24 days since I posted the question on MSE; since then I've started through MacLane/Birkhoff and the first two chapters of Basic Algebra II. However Rotman's text looks interesting and I'd love to borrow a copy from the library. Until then, could you please give a comparison between MacLane/Birkhoff and Rotman's text? I looked through the links you've shared, but none had a direct comparison between the texts I've mentioned - and this would have been most helpful. I'd be much obliged if you'd kindly compare the books for me. $\endgroup$ – Ishan Jun 18 at 17:46
  • $\begingroup$ Hi, Ishan. I updated my answer, so you could discern the content from the third edition of Rotman's Advanced Modern Algebra, Part 1 and Part 2. Given Part 2 of Rotman's new edition, I'm inclined to favor Rotman, over MacLane/Birkhoff. But, as I suggested in my post, I personally, if I were in your position at this point in time, and given your experience, I'd get both Rotman's and MacLane/Birkhoff. But given what you are currently using and with your interest. previous learning, and motivation, you'll master what you hope to. Good luck! $\endgroup$ – Namaste Jun 18 at 18:06
  • $\begingroup$ Dear @Namaste Thank you again. I agree with your advice on using both together (Rotman's being the primary text). I've just been able to confirm that the library does indeed have only the second edition of Rotman's text. I was wondering if you would still favour Advanced Modern Algebra (2e) over MacLane/Birkhoff? $\endgroup$ – Ishan Jun 18 at 18:19
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    $\begingroup$ Rotman's Modern Algebra is probably a good choice. I can learn a lot from his writing without much effort. $\endgroup$ – James S. Cook Jul 3 at 15:25
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I taught myself the basic material from the first 6 chapters of the 2nd edition of Rotman's Advanced Modern Algebra, and I loved it and felt things were explained clearly. Earlier I had tried teaching myself out of Artin's Algebra but got stuck or spent too much time trying to understand with details of the wallpaper group (which in retrospect I wish I hadn't done) and my recollection is that it postpoines galois theory for quite a bit. Rotman on the other hand gets to basic Galois theory at least by chapter 6, possibly earlier. I wish I had learned the subject out of Dummitt and Foote, as it is very comprehensive and has lots of examples. Paul Cohn's Basic Algebra was also a book I found clearly and concisely written and I taught myself things from there. I'm didn't try Lang or Jacobson, so I can't exactly answer your question.

I should also mention Prof. Anthony Knapp has made basic algebra analysis textbooks he wrote freely available, but I haven't read them but the table of contents look impressive

http://www.math.stonybrook.edu/~aknapp/

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other possibilites

Joseph Gillian - Contemporary Abstract Algebra https://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702

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    $\begingroup$ I have to disagree with this recommendation, for a person in the apparent situation of the questioner. Namely, J. Gallian's book is a quite slow-paced introduction to just the basics of abstract algebra, and... while it might be a reasonable choice for the first abstract algebra book someone sees... it has two important limitations, that should be understood: it does not treat very much abstract algebra, and what it does treat is in a fairly indulgent style, which is not what readers will encounter in any later forays into the literature. $\endgroup$ – paul garrett Jul 1 at 22:04

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