A Book for abstract Algebra I am self learning abstract algebra. I am using the book Algebra by Serge Lang. The book has different definitions for some algebraic structures. (For example, according to that book rings are defined to have multiplicative identities. Also modules are defined slightly differently....etc) Given that I like the book, is it OK to keep reading this book or should I get another one?
Thank you
 A: If you're happy with Lang's Algebra, it's a fine book to stick with.  As with any topic, you might want to take a "peek" elsewhere, for reference, alternate exercises, and/or different approaches.
I agree that Fraleigh's A First Course in Abstract Algebra is a great text, particularly when just beginning to study Abstract Algebra. It is very intuitive, it has great exercises that help build competency with proofs, and the exercises are organized according to level of difficulty, beginning, e.g., with problems that are more computational, and building to more challenging proofs. In that sense, it is very developmental in its structure.  The other thing worth noting is that it is "group" oriented.
One text that has not yet been mentioned yet is Michael Artin's Algebra, 2nd ed.. A brief overview of the text, along with links to various merchants, can be found here @Google.com. If you go to MIT's open course-ware site, you'll find more exercises and supplementary lecture notes for the Algebra class as taught by Artin at MIT Open Course Ware, which complements his text. Artin's text is particularly attractive for those who have a encountered "the basics" of linear algebra; it builds on students' exposure to linear algebra (however elementary) and connects it to algebra, as a whole. (However, one needn't have had any advanced coursework in linear algebra, since the text begins with a thorough introduction covering any material that will be relevant later in the text.)
If you'd like to supplement Lang's text, without a lot of expense, you might want to check out Beachy and Blair's website for an Abstract Algebra study guide you can download (~200 pages, pdf), and supplements, as well. Or perhaps bookmark their Online Study Guide for practice problems, and review.
A: A great book that has not been mentioned is Contemporary Abstract Algebra by Joseph A. Gallian.
I consider it to be one of the best books in mathematics. It is an extremely student friendly book and is full of excellent examples.
A: If you're self-studing, this complete set of free video lectures by Benedict Gross at Harvard is invaluable.
http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra
It follows "Artin" which I think is a great place to start as it gives a very intuitive approach to the key concepts of group theory. 
After group theory, I would would recommend a switch to "Dummit & Foote." (Mentioned several times above) It's very thorough and has many pertinent examples. (These subsequent sections - after group theory - are written by Dummit.)
A: Lang's "Algebra" is without doubt one of the classic references but sticking to it, or to only one book for that matter, depends highly on your style tastes, background level and aim. A beginner in abstract algebra may find the exercises in Lang to be too hard or/and to leave important concepts and results as problems.
You should check out other standard references listed below, and see their style to decide which suits you best:


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*Dummit; Foote - "Abstract Algebra", Wiley 2004.

*Rotman - "Advanced Modern Algebra", AMS 2011.

*Hungerford - "Algebra", Springer 1974 (2003).

*Grillet - "Abstract Algebra", Springer 2007.


The first is the standard title used in many American graduate schools for the preliminary/qualifying examination in graduate algebra, although many people would consider it an upper undergraduate book. It is filled with lots of exercises and examples. The third book is very abstract and general, encyclopedic and good as a reference since it is highly formally organized with just definitions-theorems-proofs-corollaries, with a great selection of exercises (I think better than Lang's); but, although it covers with great generality topics the others take longer to explain, it lacks chapters on homological algebra or representations, which the others have. Finally the second book is a mixture of all styles, and in particular is my favourite. It is the biggest, with around 1000 pages, it has an informal style in its explanations but a highly rigorous development, with insightful examples and filled also with lots of exercises (which I particularly find well-suited for self-study). The last one is a quite recent textbook which some people will detest while others will love.
There are many other abstract algebra titles, this is just a small standard selection. Although many consider Lang to be better, do not forget that it is always best to learn from at least two different sources, so that you can clarify dark passages of one in the other, fill complementary material or provide easier examples/exercises to start with.
Once you have mastered the standard graduate course, or if you want to deepen your knowledge on particular chapters of the previous books, like commutative algebra for example as requisite for algebraic geometry, you may jump to the next level of more specialized texts:


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*Roman - "Advanced Linear Algebra", Springer 2008.

*Fulton; Harris - "Representation Theory, A First Course", Springer 1991.

*Eisenbud - "Commutative Algebra with a View toward Algebraic Geometry", Springer.

*Singh - "Basic Commutative Algebra", World Scientific 2011.

*Matsumura - "Commutative Ring Theory", AMS 2011.

*Rotman - "An Introduction to Homological Algebra", Springer 2009.

*Weibel - "An Introduction to Homological Algebra", Cambridge U. Press 1994.

A: In addition to the other wonderful responses, you might want a very accessible and gentle introduction. 
If so, I recommend the (cheap) book:
A Book of Abstract Algebra, 2nd Edition by Charles C. Pinter (Dover Book).
Regards -A
A: Try Jacobson's Basic Algebra I, now available in a Dover edition.
A: At my university, we used John B. Fraleigh's book "A First Course in Abstract Algebra" (Amazon). It assumes very little and contains a lot of exercises.
If you a little more conceptual approach (and don't mind reading a lot), you could check out Paolo Aluffi's "Algebra: Chapter 0". (Amazon)
A: This is a specialized answer, but it is too good not to mention in my opinion.
If you want an excellent and fairly exhaustive book on group theory, I recommend Kargapolov's Fundamentals of the Theory of Groups. Great exposition and clear, succint, insightful proofs.
The only complaint I have is that there are not enough exercises. The book is so masterful that I'm left hungry for what else Kargapolov expects the reader to be fluent with.
A: There is a less famous but very nice book Abstract Algebra by Paul B. Garrett
and then there is the old book "A survey of modern algebra" by Birkhoff
A: It is not a book but if it can help there are a lot of materials (videos classes, lecture notes, book references, assignements...) provided for free at the MIT Open Course Ware Initiative web site (http://ocw.mit.edu/courses/find-by-topic/)
