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Algebra text with category theory

  1. Nov 23, 2012 #1
    I'm looking for a textbook that covers all of the standard undergrad algebra topics but from a more modern perspective. For example, most books reprove the isomorphism theorems for groups, rings, modules, instead of showing that all of these structures are universal algebras. Ideally the text would emphasize recurring themes that unify the subject and make the details follow easily instead of being arbitrary.

    Has anyone had any experience with Mac Lane & Birkhoff's book Algebra or Aluffi's book Algebra: Chapter 0?
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  3. Nov 24, 2012 #2


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    I read some of MacLane's Algebra textbook, it does discuss in the first chapters about Categories.

    I can't say more than that cause I haven't read a lot from it.

    But from what I read the reading was quite fluent I didn't feel that I don't understand what's going on (there are some books, especially written by Russians that are hard to follow, not this one).
  4. Nov 24, 2012 #3
    What about Lang?
  5. Nov 24, 2012 #4
    yeah, definitely lang although he covers a lot more than just the standard undergrad stuff.
  6. Nov 24, 2012 #5
    I have read a portion of Aluffi, and I like it. If oyu are interested you should just check out every book and see which works for you.
  7. Nov 24, 2012 #6
    I've heard that Lang is full of typos and is a very boring read. Though if boring just means elegant and terse I'd be happy to pick it up, as I enjoyed Rudin.

    Does Lang integrate category theory and universal algebra throughout the text?

    espen180, unfortunately the closest library that has these books is several hours away.
  8. Nov 24, 2012 #7
    You can probably read exerpts of the books on Google Books or amazon, at least enough to know whether you like their style.
  9. Nov 24, 2012 #8
    I don't think Lang uses universal algebra, but there is plenty of (basic) category theory in his book. In particular, I'm fond of the way Lang introduces categorical notions in tandem with more concrete algebraic concepts (i.e. discussing coproducts and then free products of groups). I haven't read much of his book, but I wouldn't call it boring at all. I have heard that there are lots of typos--just keep a watchful eye. Also: Lang tends to use examples from lots of different fields of mathematics, e.g. covering spaces when discussing Galois groups, even some Riemann surfaces if I remember correctly. I guess some people find this frustrating, and it definitely raises the level of sophistication of the book, but I think those examples make it much more interesting.
  10. Nov 24, 2012 #9


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    MacLane & Birkhoff's Algebra is the only undergraduate text that uses category theory from the start to unify the presentation of the traditional topics if undergraduate algebra (don't confuse with Birkhoff & MacLane's A Survey of Modern Algebra). It is very readable.

    There is also Lawvere's Conceptual Mathematics - A First Introduction to Categories. It does not cover the usual complement of topics in an undergraduate algebra course but rather aims to present category theory - inevitably covering much algebra.

    Lang's Undergraduate Algebra does not discuss much less employ category theory; however, Lang's Algebra (his graduate algebra text) does.
  11. Nov 25, 2012 #10
    I have Algebra, Chapter 0 and it is one of my favourite books. I own a number of Algebra texts, but it is the only upper-level one that works hard to explain the details. It manages to weave category theory in right from the beginning in a very natural way. Aluffi is a gifted teacher.
  12. Nov 25, 2012 #11
    To clarify, I was referring to Lang's graduate text.
  13. Nov 25, 2012 #12


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    notice aluffi's book takes much longer than usual to get around to basic facts of algebra, because of spending so long on the abstract setup at the beginning. this book is obviously more interested in setting up all that abstract language than at presenting algebra. some people may feel a more natural presentation is the opposite order.
  14. Nov 25, 2012 #13
    Is Lang a good text to read through and learn the material? Most of the reviews on Amazon imply that it is great as a reference but not useful as a textbook.

    I've gotten an overview of algebra from Herstein's Topics in Algebra, though I am shaky on the material in the later chapters. I've read Friedberg for linear algebra as well and have no experience with Galois theory. Would Lang be appropriate at my level to tie my knowledge together? It is hard to judge if it is as legendarily difficult as many reviewers say it is from Amazon reviews...
  15. Nov 25, 2012 #14
    if you went with lang you'd probably feel like wile e coyote fairly quickly... I think I'd stick with herstein or another one until you have more than just an overview.
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