Introduction.

The purpose of this chapter is to relate the set of ideal valuations of a finitely generated domain B over a noetherian domain A to the ideal valuations of A itself. For this purpose the description of the ideal valuations of A in terms of the completions of the localisations of A is not convenient and we therefore use a somewhat different one, namely that a valuation ≥0 on A is an ideal valuation of A if and only if there is a finitely generated extension B of A with the same field of fractions, such that v(x) ≥ 0 on B and the centre of v on B has height 1. Note that this implies that v is a Krull valuation of B by Theorem 3.24, and we could weaken the above condition by simply requiring that v be a Krull valuation of B. The proof of this criterion is obtained by putting together Theorem 4.24 and the Corollary to Lemma 6.11. This is done in the proof of the following theorem.

THEOREM 7.11. Let A be a noetherian domain, v be a valuation on the field of fractions F of A such that v(x) ≥ 0 on A. Then v is an ideal valuation of A if and only if there exists a finitely generated extension B of A, contained in F, such that v(x) ≥ 0 on B and the centrepof v on B has height 1.