TITLE: an easy example of valuation ring which is not noetherian？
QUESTION [6 upvotes]: Is there an easy example of valuation ring which is not noetherian？

REPLY [2 votes]: Construction of valuation domains of Krull dimension $>1$:
Let $O\neq K:=\mathrm{Frac}(O)$ be a valuation domain. Consider the natural map $h:O\rightarrow k$,
where $k$ is the residue field of $O$. Let $\overline{O}$ be a valuation domain
of $k$. Then $O^\prime:=h^{-1}(\overline{O})\subseteq O$ is a valuation domain
of $K$ with the following properties:

$\mathrm{Spec}(O)\subset\mathrm{Spec} (O^\prime)$,
$O=O^\prime_M$, where $M$ is the maximal ideal of $O$,
$O^\prime/M\cong\overline{O}$.

In particular: $O^\prime$ is never noetherian.