An affine algebra over a field K is simply a finitely generated commutative k-algebra. We are interested in such algebras not only because they provide a readily available fund of examples of commutative Noetherian rings (see 8.11), but also because such algebras have fundamental importance in algebraic geometry. In this book, we are not going to explore the reasons for this: the interested reader might like to study Miles Reid's book [18] to discover something about the connections.

What we are going to do in this chapter, in addition to developing the dimension theory of affine algebras over fields and linking this with transcendence degrees, is to prove some famous and fundamental theorems about such algebras, such as Hilbert's Nullstellensatz and Noether's Normalization Theorem, which are important tools in algebraic geometry. Although their significance for algebraic geometry will not be fully explored here, they have interest from an algebraic point of view, and we shall see that Noether's Normalization Theorem is a powerful tool in dimension theory.

Our first major landmark in this chapter is the Nullstellensatz. Some preparatory results are given first, and we begin with a convenient piece of terminology.

definition. Let k be a field. An affine K-algebra is a finitely generated commutative k-algebra, that is, a commutative k-algebra which is finitely generated as k-algebra (see 8.9).

Observe that an affine k-algebra as in 14.1 is a homomorphic image of a ring K[X1, …, Xn] of polynomials over K in n indeterminates X1, …, Xn, for some n ∈ ℕ, and so is automatically a commutative Noetherian ring: see 8.11.