Number of prime ideals that contain a non zero ideal

Question

In the proof of proposition (12.3) of Neukirchs Algebraic Number Theory, we use the fact that for a one-dimensional noetherian integral domain, there are only finitely many prime/maximal ideals that contain a nonzero ideal. I can't see why this is the case.

Answer 1

If you use the Lemma 3.4 of the same book (that only require the domain to be Noetherian) you have that every ring $\mathfrak{a}$ contains a finite product $\mathfrak{p}_1\mathfrak{p}_2\cdots\mathfrak{p}_k$ of prime ideals, therefore if the ring is one dimensional, every prime that contains $\mathfrak{a}$ must be one of the $\mathfrak{p}_i$.