The main purpose of this chapter is to define the Chow group of a Noetherian ring and prove several of its basic properties. The Chow group is a quotient of the free group whose generators are the prime ideals of the ring, and we devote the first section to a summary of some of the main properties of prime ideals, and in particular the prime ideals associated to a finitely generated module. Much of the material in the first section can be found in several books on commutative algebra, such as Matsumura or Atiyah and Macdonald. While we prove some of these basic facts, we use others without proof, giving a reference to a place in one of these books where a proof can be found. Most of the results of Section 2, in a more general setting, can be found in Fulton.

Prime Ideals in Noetherian Rings

All rings will be assumed to be commutative and have an identity element. A module M over a commutative ring A is Noetherian if it has the ascending chain condition on submodules, or, equivalently, if every submodule is finitely generated. The ring A is Noetherian if it is Noetherian as an A-module, which means that it has the ascending chain condition on ideals or that every ideal is finitely generated.