There's this exercise in Neukirch, chapter I, §3 (i've changed the statement to deal only with the case that bothers me):
Let $\mathfrak o$ be a Dedekind domain and $\mathfrak m$ be a nonzero integral ideal of $\mathfrak o$. Prove that there exists an integral ideal $\mathfrak a$ of $\mathfrak o$ that belongs to the same class of $\mathfrak m$ (in the classes of the ideal class group) and that is relatively prime to $\mathfrak m$.
But there's this case I cannot deal with: let $\mathfrak o$ have only finitely many prime ideals and let $\mathfrak m$ be the product of all those prime ideals. Is there even an integral ideal of $\mathfrak o$ coprime to $\mathfrak m$? Thank you.
EDIT: for completeness, I add the unchanged statement of the exercise:
"Exercise 8. Let $\mathfrak m$ be a nonzero integral ideal of the Dedekind domain $\mathfrak o$. Show that in every ideal class of $Cl_K$, there exists an integral ideal prime to $\mathfrak m$."
You're right, there's no such ideal, since an ideal $\neq\mathfrak o$ is contained in a maximal ideal, and this maximal ideal contains $\mathfrak m$, by the definition of $\mathfrak m$.
Note: A Dedekind domain with a frinite number of maximal ideals is a P.I.D.