[Math] $\mathbb{Z}$ is Noetherian but not Artinian

Question

I do believe that my question is simple. I do not understand what is wrong. So help me sort it out please. I know the following facts:

$\mathbb{Z}$ is a Noetherian ring and it is not Artinian because the infinite sequence $(\mathbb{Z}/2\mathbb{Z}) \supseteq (\mathbb{Z}/4\mathbb{Z}) \supseteq (\mathbb{Z}/8\mathbb{Z}) \cdots $ doesn't hold the Descending Chain Condition.

And

A ring $R$ is Artinian iff $R$ is Noetherian and every prime ideal is maximal.

We see that all prime ideals have the form $p\mathbb{Z}$ and are maximal. This is example of a module which is Noetherian but not Artinian as well.

Answer 1

$0$ is a prime ideal of $\mathbb{Z}$ that is not maximal.