In Chapter 14, we studied the highly satisfactory dimension theory for finitely generated commutative algebras over fields. Of course, finitely generated commutative algebras over fields form a subclass of the class of commutative Noetherian rings: in this chapter, we are going to study heights of prime ideals in a general commutative Noetherian ring R, and the dimension theory of such a ring.

The starting point will be KrulPs Principal Ideal Theorem: this states that, if a ∈ R is a non-unit of R and P ∈ Spec(R) is a minimal prime ideal of the principal ideal aR (see 8.17), then ht P ≥ 1. From this, we are able to go on to prove the Generalized Principal Ideal Theorem, which shows that, if I is a proper ideal of R which can be generated by n elements, then ht P ≥ n for every minimal prime ideal P of I. A consequence is that each Q ∈ Spec(R) has finite height, because Q is, of course, a minimal prime ideal of itself and every ideal of R is finitely generated!

There are consequences for local rings: if (R, M) is a local ring (recall from 8.26 that, in our terminology, a local ring is a commutative Noetherian ring which has exactly one maximal ideal), then dim R = ht M by 14.18(iv), and so R has finite dimension. In fact, we shall see that dim R is the least integer i ∈ ℕ0 for which there exists an M-primary ideal that can be generated by i elements.