Exercise 13, § 1

Let $A$ be an integral domain and $(p_i)_{i \in I}$ a family of non-zero elements such that the principal ideals $Ap_i$ are prime; let $S$ be the multiplicative subset, generated by the $p_i$.

(a) Show that $A = S^{-1}A \cap (\bigcap_i A_{Ap_i})$.

(b) Suppose that every non-empty family of principal ideals of $A$ has a maximal element. Show that each of the rings $A_{Ap_i}$ is a discrete valuation ring (cf. Chapter VI, § 3, no. 5, Proposition 9). Deduce that, if $S^{-1}A$ is integrally closed (resp. completely integrally closed), $A$ is integrally closed (resp. completely integrally closed). If $S^{-1}A$ is a Krull domain, show that $A$ is a Krull domain.