Lemma 2

Let $(p_i)_{1 \leq i \leq k}$ be a non-empty finite family of prime ideals of $A$ of height 1 and let $S = \bigcap_i (A - p_i)$; then the ring $S^{-1}A$ is a principal ideal domain.

$S^{-1}A$ is a semi-local ring whose maximal ideals are the $m_i = p_i S^{-1}A$ for $1 \leq i \leq k$, the local ring $(S^{-1}A)_{m_i}$ being isomorphic to $A$, (Chapter 11, § 3, no. 5, Proposition 17) and hence a discrete valuation ring. The ring $S^{-1}A$ is therefore a Dedekind domain (§ 2, no. 2, Theorem 1 (f) and, as it is semi-local it is a principal ideal domain (§ 2, no. 2, Proposition 1).