Another Error in Neukirch's Algebraic Number Theory?

I'm reading Neukirch's Algebraic Number Theory and trying to do the exercises. I think I may have found another error, but am not sure...

Exercise 7. In a noetherian ring $R$ in which every prime ideal is maximal, each descending chain of ideals $a_1 \supseteq a_2 \supseteq \ldots$ becomes stationary

Hint: Show as in (3.4) that (0) is a product $p_1 \ldots p_r$ of prime ideals and that the chain $R \supseteq p_1 \supseteq p_1p_2 \supseteq \ldots \supseteq p_1\ldots p_r = (0)$ can be refined into a composition series.

So, my main issue is that the ring $\mathbb{Z}$ is noetherian and every prime ideal is maximal, yet the chain $$(2) \supsetneq (4) \supsetneq (8) \supsetneq \ldots \supsetneq (2^n) \supsetneq \ldots$$ Never becomes stationary.

Additionally, I don't know what (correct) statement he could be asking me to prove. (I also realize there may well be a translational error.) So am I reading it wrong or what?

The (0) ideal in $\mathbb{Z}$ is prime, but not maximal. So $\mathbb{Z}$ does not satisfy the hypotheses.