I have a question regarding some Commutative Algebra facts. I am using the book A Course in Commutative Algebra by Gregor Kemper. On page 24 he states that if $k$ is a field, the polynomial ring $k[x]$ is Noetherian, and not Artinian. And in theorem 2.8 he states:
Let $R$ be a ring. Then the following statements are equivalent:
$R$ is Artinian $\Leftrightarrow$ $R$ is Noetherian and every prime ideal of $R$ is maximal.
Now I learned in my basic algebra courses that if $k$ is a field, then $k[x]$ is a principal ideal domain. And in a PID, every prime ideal is maximal. But this would be in contradiction with theorem 2.8, since $k[x]$ is Noetherian and PID and therefore $k[x]$ must be Artinian. What goes wrong here?