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.
Best Answer
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.