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I'm very interested in learning abstract algebra and category theory on my own. It seems a very powerful tool in math and it seems worthwile to take a time and learn about it. I just don't know even where to begin. Can someone point out for me what are good references to self-study those topics ? I'm really beginner, the only thing connected to algebra that I'm familiar with is linear algebra.

Thanks very much in advance.

Edit: Until now I've studied analytic geometry, single variable calculus, multivariable calculus, linear algebra, ordinary differential equations and I'm currently studying differential geometry and multilinear algebra.

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  • $\begingroup$ To get a useful answer, you should probably give us a better idea of your background. $\endgroup$
    – Chris Godsil
    Mar 15, 2013 at 21:55
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    $\begingroup$ Please don't use Lang's Algebra. I'm shocked that people are recommending it. I literally could not think of a worse book for a self-studying beginner with no background. It's like asking to learn English and being handed the dictionary. $\endgroup$
    – Alexander Gruber ♦
    Mar 15, 2013 at 22:10
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    $\begingroup$ If you like lecture videos, I can recommend these free videos for an introduction to abstract algebra. $\endgroup$
    – hammar
    Mar 15, 2013 at 23:43
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    $\begingroup$ You should consider learning some algebraic topology before you start thinking about category theory. $\endgroup$
    – user314
    Mar 16, 2013 at 2:50
  • $\begingroup$ I very much enjoyed Category Theory in Context, by Emily Riehl. I started reading it while I was being introduced to differential geometry (of manifolds) and after reading about the definition of the tensor product in terms of its universal property. For me it was a convenient moment to get into categories. $\endgroup$
    – Elías Guisado Villalgordo
    Sep 21 at 8:53

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I would really recommend that you plunge in and seriously read Aluffi's Algebra - chapter 0. The book will cover many many topics in abstract algebra including group theory, ring theory, field theory, as well as more advanced material like homological algebra.

The book is marvelously written which is a reason on its own for reading it. But, since you are looking for the category theory perspective this book is really what you are looking for. It does not assume any category theory, but instead develops parts of at as you go along, exemplifying everything with the algebra being developed at the same time.

Depending on you level of comfort with abstract ideas, you might find that you want to reinforce reading the book with reading a more elementary text on group theory. Rotman's Group Theory is excellent.

I really don't like Fraleigh's book, though I know it's popular. In my opinion the order in which things are presented makes little categorical sense.

The nice thing about Aluffi's book is that when you finish it you can truly say that you know the chapter 0 of modern algebra. It really gives you a very sound foundation of all of modern algebra.

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Given your background, it seems to me that Fraleigh's A First Course in Abstract Algebra would be a great start for a beginner. I've had much success with it in terms of its use in an undergraduate's first course in abstract algebra/modern algebra.

Start with abstract algebra. As Alexander suggested, it wouldn't hurt to work through Fraleigh in conjunction with Dummit and Foote. Then when you get a good "lay of the land", incorporate/begin study in category theory.

See this post: When to learn Category Theory?

See this post: For more suggestions on abstract algebra texts, at variying levels of difficulty.

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Here are a couple of free documents that have been created relatively recently that look quite good (disclaimer: I haven't finished either).

Algebra Chapter 0 - Aluffi

Category theory for scientists - Spivak

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I would pick up Fraleigh and Dummit and Foote, read them congruently, and forget about category theory until you're done with both of them.

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As a self-learner, I have found it useful to learn about category in the context of topology. For example, Rotman's "An Introduction to Algebraic Topology" frames ideas in terms of categories from the beginning. This might give a better motivation than seeing categories for the first time in a purely algebraic context.

Edit: Just looked back and saw your background, and category theory will probably not make much sense to you yet. I won't get rid of my post because it remains a good strategy after you have learned some basic abstract algebra. Category theory one more level of abstraction beyond what you typically learn in your first abstract algebra class.

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S.Lang, Algebra

S.Mac Lane, Categories for the Working Mathematician

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  • $\begingroup$ Not for the faint of heart :) $\endgroup$
    – Shaun Ault
    Mar 15, 2013 at 22:28
  • $\begingroup$ @Shaun Ault: Let us hope that user1620696 is not faint! $\endgroup$
    – Boris Novikov
    Mar 15, 2013 at 22:38
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Read Chapter 5.10 from Teach Yourself Logic (pdf) for guidance about learning category theory. Category theory has as its prerequisites not only abstract algebra, but also advanced logic.

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Good texts for abstract algebra self-study are ones by Herstein, and Pinter. The Pinter book is the easier of the two and it is available from Dover so it won't cost too much. There are a lot of good online sources available too, you just have to look. As far as category theory goes, it might be better to get a firm grounding in algebra first since it is an essential component of category theory. Hope this helps. Good luck with the study!

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If you are really serious about diving into the deep, then i recommend Serge Lang's "Algebra". Unlike many other textbooks on Abstract Algebra, Lang brings in category theory from the beginning of his development. Strictly speaking, no background is necessary to understand the abstract theorems, other than linear algebra. However, it will require considerable effort from your part to fill in the various gaps in the proofs that occur very often. Check out some reviews of that book in Amazon, you will get a good idea of what it is about. Ideally, you could also complement your study with a copy of Dummit and Foote, which is much easier book to read, and which you could use every time you get stuck with Lang.

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