Proposition 8

Let $A$ be an integrally closed Noetherian domain. (a) Let $P$ be an extremal divisor of $A$ and $p$ the corresponding divisorial prime ideal;

for $n \in \mathbf{N}$, let $p^{(n)} = p^n A_p \cap A$; then $p^{(n)}$ is the set of $x \in A$ such that $v_p(x) \geq n$ and is a $p$-primary ideal.

(b) Let $a$ be a divisorial integral ideal, $n_1 P_1 + \ldots + n_r P_r$ the divisor of $a$ (the $P_i$ being distinct extremal divisors) and $p_i$ the divisorial prime ideal corresponding to $P_i$. Then $a = \bigcap_{i=1}^r p_i^{(n_i)}$ is the unique reduced primary decomposition of $a$ and the $p_i$ are not immersed.

By Corollary 1 to Proposition 6, the relation $x \in p^n A_p = (pA_p)^n$ is equivalent to $v_p(x) \geq n$; on the other hand, as $A$, is a discrete valuation ring, $(pA_p)^n$ is $(pA_p)$-primary (Chapter IV, § 2, no. 1, Example 4) and hence $p^{(n)}$ is $p$-primary (Chapter IV, § 2, no. 1, Proposition 3); this shows (a). Proposition 5 certainly shows that $a = \bigcap_{i=1}^r p_i^{(n_i)}$. As $p_i \not\subset p_j$ for $i \neq j$ this primary decomposition is reduced: For if $p_i^{(n_i)} \supset \bigcap_{j \neq i} p_j^{(n_j)} \supset \prod_{j \neq i} p_j^{(n_j)}$, $p_{ji}$ would contain one of the $p_j$ for $j \neq i$ (Chapter II, § 1, no. 1, Proposition 1). The uniqueness follows from Chapter IV, § 2, no. 3, Proposition 5.