Proposition 1

A semi-local Dedekind domain is a principal ideal domain.

Let $A$ be a semi-local Dedekind domain, $K$ its field of fractions, $p_1, \ldots, p_n$ its maximal ideals and $v_1, \ldots, v$, the corresponding essential valuations; these are the only essential valuations of $A$. Let $a$ be a non-zero integral ideal of $A$. Since it is divisorial, there exists ($§ 1$, no. 4, Proposition 5) integers $q_1, \ldots, q_n$ such that $a$ is the set of $x \in K$ such that $v_i(x) \geq q_i$ for $1 \leq i \leq n$. Let $x_0$ be an element of $K$ such that $v_i(x_0) = q_i$ for $1 \leq i \leq n$ (Chapter VI, $§ 7$, no. 2, Corollary 1 to Theorem 1). Then $a$ is the set of $x \in K$ such that $v_i(xx_0^{-1}) \geq 0$ for $1 \leq i \leq n$. Thus $a = Ax_0$.

If $A$ is a Dedekind domain, it has been seen, in the proof of Theorem 1, that the group D(A) of divisors of A is identified with the group I(A) of fractional ideals $a \neq (0)$ (as A is Noetherian, every non-zero fractional ideal is finitely generated). The divisor class group C(A) of A ($§ 1$, no. 2) is then identified with the group of classes of ideals $\neq 0$ of A (defined in Chapter II, $§ 5$, no. 7).