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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.

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 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 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). 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 few portions from Basic Algebra II (such as Group Representation Theory) were best done from elsewhere.

I would be grateful if somebody could compare using Basic Algebra I, II , MacLane and Birkhoff’s Algebra , and Lang’s Algebra; In particular, 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!

(Syllabus below)

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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    $\begingroup$ “I would prefer to learn Abstract Algebra using Category Theory openly” — you should consider picking up Aluffi’s Algebra: Chapter 0, although I know it’s not on your list. I love this book. It has a great tone, a good perspective, and was helpful in understanding category theory via algebra. $\endgroup$ Commented May 25, 2019 at 13:18
  • $\begingroup$ See math.stackexchange.com/a/832845/589 $\endgroup$
    – lhf
    Commented May 25, 2019 at 15:47
  • $\begingroup$ I don't know enough about all of the mentioned texts, but I learned algebra out of Lang and was very happy with it. There are loads of nice exercises. I have heard people say it is better as a "reference book" than a "learning book" since it has a very wide scope, and I agree to some extent, but I had no major complaints with it. $\endgroup$
    – TomGrubb
    Commented May 27, 2019 at 4:22
  • $\begingroup$ @Ishan It's considered good manners to notify the reader when that is the case, in order to avoid answering questions that have already been answered elsewhere. $\endgroup$ Commented May 27, 2019 at 6:45
  • $\begingroup$ @Ishan I took the liberty of editing this one. Do feel free to edit it and then post the same disclaimer on MO. $\endgroup$ Commented May 27, 2019 at 6:49

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