TITLE: Example: Krull dimension 1 but not a PID
QUESTION [9 upvotes]: It's easy to prove that if $A$ is a PID which is not a field then $\dim A= 1$. What is a counterexample to the converse? Thanks for any insight.

REPLY [7 votes]: $\mathbb Z[\sqrt{-n}]$, for $n\ge 3$ and square-free, is one-dimensional (why?) and it's not a UFD.