Descending chain of ideals becoming stationary

Question

Exercise 3.7 of Algebraic Number Theory (Neukrich) is:

In a noetherian ring R in which every prime ideal is maximal, each descending chian of ideals $\mathfrak{a_1 \supset a_2 \dots}$ becomes stationary.

Isn't $\mathbb Z$ with $\mathfrak {a_i} = (p^i)$ a counterexample?

Answer 1

But $(0)$ is a prime ideal in $\mathbb{Z}$ while it is not maximal since $(0) \subsetneq (p)$ for any prime $p \geq 2$. The ring really needs to be dimension $0$ for Noetherian to imply Artinian. All domains which are $0$ dimensional must be fields since $(0)$ has to be maximal.