It's easy to see that $\mathbb{Z}$ is noetherian but not artinian. In my course notes there's proven: $A$ a commutative ring: $A$ artininian $\Leftrightarrow$ $A$ noetherian and Spec($A$) = Max($A$)
with Spec($A$) = collection of prime ideals,
so there must be a prime ideal in $\mathbb{Z}$ that is not maximal. Can someone give me an example of this?
