Reading commutative algebra book Now I am just a beginner in commutative algebra, so I just want to ask which book I should read step by step. I am reading Step in commutative algebra of Sharp, then I want to read Commutative ring theory of Matsumura before reading Introduction to commutative algebra of Antiyah and Macdonald because I was suggested that this book is very difficult and I need to learn it in the long time. So can you give me any suggestion?
 A: If you have studied basic ring and module theory, for example, from Dummit & Foote, my suggestion is that you can start reading and solving exerices in Atiyah & Macdonald. In fact, most universities teach commutaive algebra from this book, and you can find solutions to most exercises online, as supplement to your self-study. This book is not hard, but very dense. If you are stuck with some concept, you can refer to Dummit & Foote, chapter 7,8,10,15, and 16. After your study, you can check out this problem set:
http://www.math.ucla.edu/~merkurev/215a.1.16f/problems.pdf
and see if you can solve them.
Good luck!
Update: if you are not sure which problems are more important, check out this page:http://www2.gsu.edu/~matyxy/math831/math831.html. The problems assigned are very typical(those you need to know) and it comes with nice solutions.
A: I can recommend the following books:
Introduction to Commutative Algebra by Atiyah and Macdonald: A classic book with a terse treatment of the material and many insightful exercises. Highly recommended as a primer.
A Course in Commutative Algebra by Kemper: A more recent book with a strong emphasis on the Hilbert Nullstellensatz, which allows geometric interpreation of commutative algebra. Both concise and with ample geometric motivation. Has some material on computational aspects, but doesn't cover some standard material.
Commutative Algebra with a view towards Algebraic Geometry by Eisenbud is a massive tome with a wealth of material and motivation. Don't be intimidated by the sheer size of it, you don't need to read it in its entiretiy to get a good grasp on commutative algebra. Also has an extensive appendix on homological algebra.
Commutative Ring Theory by Matsumura: Goes deeper than Atiyah/Macdonald, but is also written quite concisely. Covers most of the material that Eisenbud does, except for the computational stuff, but has an approach that more heavily uses universal properties, whereas Eisenbud seems to favour concrete construction. Less geometric motivation than other texts such as Kemper and Eisenbud.
I also want to mention Pete L. Clark's excellent lecture notes, which cover some material not covered in the other references.
A: As a first introduction I really love the following books: 
Undergraduate Commutative Algebra (Reid)
Undergraduate Algebraic Geometry (Reid) 
(the above can be read as a pair)
Computational Algebraic Geometry (Schenck) 
And Ideals Varieties and Algorithms (Cox Little and O'Shea) (if you only want to read one book, you could read this as an introduction.)
