I would like to find an algebra book that would suit my needs. It should be advanced graduate/graduate level. I have some experience in group and ring theory but zero in linear algebra.

I'm not scared of graduate books IF there is no prerequisite material on linear algebra and so on(some books are meant as graduate in a sense that they require some mathematical maturity, others assume some undergraduate topics, the latter wouldn't suit me ).

So, my ''ideal'' book would:

1) Go for groups $\to$ rings $\to$ modules $\to$ vector spaces or groups $\to$ rings $\to$ fields $\to$ modules/vector spaces sequences.

2) Not be too long. Well, I don't ask for 200-300 pages text, but something like 500-600. That said, it is desired, but not essential, if the book has other advantages.

3) I'd like it to be modern, that is, released within 20 years before now. But this is not highly essential for me.

4) The book might introduce some category theory language. But it shouldn't cloud the main topic's exposition.

I've started studying Algebra with Aluffi's book, which is a really good book, but, of course, have some flaws. Now I want to try something else, something a little different.

I heard Grillet wrote a really good book, but as far as I'm concerned, it assumes knowledge of linear algebra (which I don't want to learn outside modules/field/ring theory).

I suppose Rotman's Advanced Modern Algebra might suit me, but it's a little too long and has too many topics (which I don't intend to learn just now).

That said, ANY advice would be HIGHLY appreciated. Even if u just came to recommend something that might not suit the conditions above, please, feel free to say, if you think it is worth mentioning (but, please, tell, why do you think a book is so great).

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    $\begingroup$ I personally like Serge Lang's Algebra for graduate students, it is quite long, but has nice chapters about linear algebra and homological algebra. $\endgroup$
    – Lullaby
    Oct 23 '15 at 21:47
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    $\begingroup$ + Lang's Algebra... Pretty much it's the Algebra bible. Lang doesn't assume you have any exposure to anything but he assumes you have some math maturity. Sometimes he says something is "obvious" but it may take a few minutes to work out why it is true if you're a beginner. But it pretty much contains EVERYTHING. $\endgroup$
    – Hamed
    Oct 23 '15 at 21:53
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    $\begingroup$ agree S. Lang's is the book! $\endgroup$
    – janmarqz
    Oct 23 '15 at 21:54
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    $\begingroup$ @MorganRodgers I think the uniqueness of this question is in asking for no-linear-algebra book. That's why the highest upvoted answer is not a book recomendation, but an explanation about why that's a bad idea. $\endgroup$
    – 5xum
    Oct 24 '15 at 16:05

This was first meant as a comment but got rather long, so I am posting it as an answer.

Trying to learn abstract algebra without linear algebra is sort of like trying to dig a hole without a shovel. It's possible, sure, but you'll have plenty of moments when you'll go "man, I wish I had a shovel".

Even worse, when you ask other people to show you how to dig the hole, they will most likely say "Well, first you take a shovel and... Wait, what do you mean you don't have a shovel? But digging a hole without a shovel, that's like trying to learn abstract algebra without learning linear algebra first!"

OK, maybe I got carried away with the metaphor, but my point is that linear algebra will introduce you to many tools that are useful even in more abstract algebraic applications.

For example, when learning about rings, it's often useful to know a lot about matrices and how they work. They allow you to have a ready made set of examples for rings, which you can then use if you, for example, want to produce a counterexample to some theorem.

I would highly recommend you learn at least the basics of linear algebra (vector spaces, matrices, diagonalization...) as it gives you a fundamental knowledge that is useful in abstract algebra, multivariate analysis, differential equations, numerical mathematics, probability theory, statistics... basically any field you can think of will require at least the basic knowledge of linear algebra.

  • $\begingroup$ Loved the metaphor.... Thumbs up. $\endgroup$
    – Hamed
    Oct 23 '15 at 21:59
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    $\begingroup$ @Hamed Thanks, I was feeling creative. Hopefully, it also got the point across. I may be biased toward linear algebra, but I think it's so fundamental that going into any advanced topic without it is suicide. $\endgroup$
    – 5xum
    Oct 23 '15 at 22:01
  • $\begingroup$ I agree 100 percent. $\endgroup$
    – Hamed
    Oct 23 '15 at 22:02
  • $\begingroup$ I couldn't agree more. Instead of an abstract algebra book, maybe Linear algebra by Fraleigh et al. would be a better place to start. $\endgroup$
    – David Hill
    Oct 24 '15 at 0:12
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    $\begingroup$ I will add this link, which seems related: Why teach linear algebra before abstract algebra? $\endgroup$ Oct 24 '15 at 6:01

Lang is a great book, but might be a bit tough to learn from depending on your background. So I will suggest one that I really enjoyed learning from as an independent study in college. The book by Dummit and Foote. It is a bit long for what you describe, but a nice beginning text to get started into graduate level algebra. The length is mostly due to them having so many examples and exercises. I found that to be wonderful for learning algebra. You can always skip examples/exercises if you have had enough. If you find it is not deep enough, there are quite a few interesting and more recent topics that are added in at the end of the book. Things like homological algebra, representation theory, category theory, algebraic geometry, commutative algebra.

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    $\begingroup$ I second Dummit and Foote. I'd recommend googling it for reasons I don't want to spell out explicitly. $\endgroup$
    – David Hill
    Oct 24 '15 at 0:11

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