Textbook in algebra leading to research

Question

I am searching for a book in commutative algebra with will develop theory from an undergraduate level and lead to areas of current research. Any help is welcome.

Answer 1

You can find 'A Term of Commutative Algebra' by Allen Altman and Steven Kleiman (introduces Category Theory and has full solutions) which is a more in depth and updated version of 'Introduction to Commutative Algebra' by Atiyah, Macdonald. After that you could try Commutative Algebra I/II by Zariski, Samuel. Whilst simultaneously getting acquainted with Algebraic Geometry and Homological Algebra. If you think you can work through a lot by yourself and don't need solutions, skip the first book I mentioned and just start with Atiyah's book.

These books are just a basic introduction though; but they should keep you out of trouble for a while!

Answer 2

Pinter's A Book of Abstract Algebra (Second Edition) is quite good.