I'm trying to show that
Let $\frak{a},\frak{b}$ be two ideals of a Dedekind domain $\cal{O}$. Show that there is an isomorphism \begin{equation*} \frak{a}\oplus\frak{b}\cong\cal{O}\oplus\frak{a}\frak{b}. \end{equation*}
If $\frak{a},\frak{b}$ are relative prime, then the thing is easy. For the general case, there exists some $c\in K$ such that $c\frak{a}$ is relatively prime to $\frak{b}$. Therefore \begin{equation*} c\frak{a}\oplus\frak{b}\cong\cal{O}\oplus c\frak{a}\frak{b}. \end{equation*}
Now, how to prove $\frak{a}\oplus\frak{b}\cong\cal{O}\oplus\frak{a}\frak{b}$ from this?
Eidt: I'm so fool to ignore that $c\frak{a}\cong\frak{a}$. So there is no problem now.
