For personal interest, I've been working through the exercises in Algebra(by Serge Lang) to create my own solution. Then I came across an exercise on Dedekind rings.
The exercise is as follows:
Prove that every ideal of Dedekind ring $\mathfrak o$ is finitely generated.
with hint:
Given an ideal $\mathfrak a$, let $\mathfrak b$ be the fractional ideal such that $\mathfrak a\mathfrak b=\mathfrak o$. Write $1=\sum a_ib_i$ with $a_i\in\mathfrak a$ and $b_i\in\mathfrak b$. Show that $\mathfrak a=(a_1,\cdots,a_n)$.
Every part except existence of $\mathfrak b$ is done. But I cannot justify that $\mathfrak b$ exists. First I tried as $\mathfrak b$ be colon ideal $(\mathfrak o:\mathfrak a)$, but I cannot justify existence of non-zero $c\in\mathfrak o$ such that $c\mathfrak b\subset\mathfrak o$.
How can I construct $\mathfrak b$? Any help is welcomed.
I want to prove this statement with definition of Lang(fractional ideals form a group), so can you avoid alternative definitions such as factorization of nonzero ideals or Krull dimension?