In comments, Grétar Amazeen asks:
Is Langs Algebra a good book? I just got it in the mail and I´m going to use it to brush up on my algebra before I start graduate school. I´ve heard that he uses his own private nomenclature, is that something I´ll have any problems with?
Michael has already given word-for-word my answer to the question:
Oh dear god no.
Lang does use his own private nomenclature (“entire rings”, for example), but that’s a minor issue. The book is just hard to read. The only chapter that I thought was well-written was the group theory chapter, but it’s very concise, so it might not be good for your purposes.
Abstract algebra has two excellent textbooks that are pitched at the advanced undergraduate or introductory graduate level: I. N. Herstein’s Topics in Algebra, and Michael Artin’s Algebra. Herstein covers the standard topics very clearly. Artin gives a much broader introduction to algebra’s relationship to other fields of mathematics, so it’s good for inspiration.
A few topics not covered in Herstein that are worth knowing are:
- The Nullstellensatz, and the relationship between algebraic varieties and ideals of commutative algebras.
- The theory of semisimple algebras, the Wedderburn-Artin Theorem, and its applications, such as Maschke’s Theorem for group representations.
(These are probably all covered in Artin, but I don’t have my copy handy so I’m not completely sure. They are all covered in Lang, but in both cases the chapters aren’t very good.)
A more idiosyncratic suggestion I have is Ideals, Varieties, and Algorithms, by Cox, Little, and O’Shea. It covers the Nullstellensatz, but from the point of view of Gröbner bases, which are a computational tool that makes it easy to work out examples in commutative algebra. They also make it easier to understand why homological algebra is interesting from an algebraic point of view, and not just as a tool in algebraic topology, again because they make examples easy to work out.