Definition 2

Every element a of I(A) such that $a = \tilde{a}$ is called a divisorial fractional ideal of A.

In other words a divisorial ideal is just a non-zero intersection of a non-empty family of fractional principal ideals. Every non-zero intersection of divisorial ideals is a divisorial ideal. If a is divisorial, so is ax for all $x \in K^*$, the mapping $b \mapsto bx$ being a bijection of the set of fractional principal ideals onto itself, For all $a \in I(A)$, 6 is the least divisorial ideal containing a and is equivalent to a modulo R. Moreover, if $b$ is a divisorial ideal equivalent to a modulo R, then $\tilde{a} = 6 = b$. Hence 5 is the unique divisorial ideal $b$ such that div $a =$ div $b$ (in other words, the restriction of the mapping $a \mapsto$ div $a$ to the set of divisorial ideals is injective).

Let a and b be two fractional ideals of K. Recall (Chapter I, § 2, no. 10) that $b : a$ denotes the set of $x \in K$ such that $xa \subset b$; this is obviously an A-module; if $b \in I(A)$ and $a \in I(A)$, then $b : a \in I(A)$; for if $d$ is a non-zero element of A such that $db \subset A$ and $da \subset A$ and $a$ is a non-zero element of $A \cap a$, then $da(b : a) \subset A$; on the other hand, if $b \neq 0$ belongs to $b$, then $bda \subset b$, hence $bd \in b : a$ and $b : a \neq 0$.

The definition of $b : a$ can also be written: