Exercise 2, § 2

(a) Let $a, b$ be two integral ideals of a Dedekind domain $A$; show that there exists an integral ideal $c$ which is prime to $b$ and such that $ac$ is principal (use Proposition 9 of § 1, no. 5).

(b) Let $a, b$ be two integral ideals of $A$; show that there exists $x \neq 0$ in the field of fractions of $A$ such that $xa$ is an integral ideal of $A$ which is prime to $b$ (same method). Deduce that the module $a/ab$ is isomorphic to $A/b$.