Exercise 9, § 4

Let $A$ be a Laskerian completely integrally closed domain (Chapter IV, § 2, Exercise 23). Suppose further, that, for all $x \neq 0$ in $A$, the prime ideals $p_i$,

(*) There exist examples of local integral domains of dimension 1 which are completely integrally closed but are not valuation rings (cf. P. Ribenboim, Sur une note de Nagata relative à un probléme de Krull, Math. Zeitsch., v. LXIV (1956), pp. 159–168).

weakly associated with $A/A_x$ (Chapter IV, § 1, Exercise 17) are all of height 1, that is to say that for all $i$, $p_i$ contains no prime ideal other than itself and 0 (cf. Chapter VII, § 1, no. 6). Show that under these conditions $A$ is an intersection of valuation rings of height 1. (Show first that $A$ is the intersection of all the rings $A_p$, where $p$ runs the set of prime ideals of height 1, using Exercise 17 (i) of Chapter IV, § 1. Then prove that these local rings $A_p$ are completely integrally closed, using condition (LA$_1$) of Chapter IV, § 2, Exercise 23. Finally apply Exercise 8.) The above hypotheses are in particular satisfied when $A$ is a Laskerian completely integrally closed domain in which every ideal admits a single primary decomposition (Chapter IV, § 2, Exercise 26).