Theorem 4

Let A be an integral domain and M the set of its prime ideals of height 1. For A to be a Krull domain, it is necessary and sufficient that the following Properties are satisfied: (i) For all p ∈ M, A, is a discrete valuation ring. (ii) A is the intersection of the A_p for p ∈ M. (iii) For all x ≠ 0 in A, there exists only a finite number of ideals p ∈ M such that x ∈ p. Moreover, the valuations corresponding to the A_p for p ∈ M are the essential valuations of A.

The conditions are trivially sufficient. Their necessity follows immediately from Theorem 3 of no. 4, Corollary I to Proposition 6 and the fact that the essential valuations of A satisfy the conditions of Definition 3 of no. 3.