TITLE: ideals with fixed norm in a Dedekind domain
QUESTION [6 upvotes]: Given a general Dedekind domain $R$ is it true that there are at most finitely many prime ideals of $R$ with the same given norm? By "norm" I mean the index of the ideal in $R$, assumed to be finite.

REPLY [5 votes]: One has the following general result:

Theorem (Gilmer--Heinzer): Let $R$ be a Noetherian ring. Then, there exists only finitely many ideals $I$ of $R$ such that $|R/I|\leqslant n$ for any
natural number $n$.

For a proof one can see [1]. For a more leisurely discussion see [2, Pg. 15].
[1] Gilmer, R. and Heinzer, W., 1992. Products of commutative rings and zero-dimensionality. Transactions of the American Mathematical Society, 331(2), pp.663-680.
[2] Anderson, D.F. and Dobbs, D. eds., 1995. Zero-dimensional commutative rings (Vol. 171). CRC Press.