Proposition 4

Let $a, b$ be two divisorial fractional ideals of $A$. The following properties are equivalent: (a) $\text{div } a$ and $\text{div } b$ are equivalent divisors; (b) there exists $x \in K^*$ such that $b = xa$.

If $\operatorname{div} b = \operatorname{div} a + \operatorname{div}(x)$ for some $x \in K^*$, then $\operatorname{div} b = \operatorname{div}(xa)$ and, as $b$ and $xa$ are divisorial, $b = xa$, which proves the proposition.

Let $a$ be an invertible fractional ideal (Chapter II, § 5, no. 6); then $a = A : (A : a)$ (loc. cit., Proposition 10) and hence $a$ is divisorial (no. 1, Proposition 1). The group $J(A)$ of invertible fractional ideals is therefore identified with a subgroup of the monoid $D(A)$ and the canonical image of $J(A)$ in $D(A)/S$ with the group of classes of projective $A$-modules of rank 1 (Chapter II, § 5, no. 7, Corollary 2 to Proposition 12 and Remark 1).