an easy example of valuation ring which is not noetherian？ Is there an easy example of valuation ring which is not noetherian？
 A: Another good example inside the field of rational functions in two variables goes like this: Choose an irrational positive number $\alpha$, and look at all rational functions $R(x,y)=\frac{P(x,y)}{Q(x,y)}$ such that when $R(x,x^\alpha)$ is written out as a formal linear combination of powers of $x$ there are no negative powers occurring.
A: Take a finite prime of the algebraic closure A of Q and complete the ring of integers of A with respect to this prime. 
A: Consider the subring $A$ of $\Bbb Q_p(X)$ consisting of rational functions defined at $X=0$ and such that $f(0)\in\Bbb Z_p$. In other words, let $B$ denote the localization of the ring $\Bbb Q_p[X]$ at the maximal ideal $(X)$ and set $A=\Bbb Z_p+XB$. It is a two-dimensional valuation ring which is therefore not noetherian (cf. Damian Rössler's comment above).
A: 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.
A: The valuation ring of $\mathbb{C}_p$ is not noetherian.
A: For a very explicit example, consider the ring
$$k[x, y, x/y, x/y^2, x/y^3, \dots]$$
localized at the origin (ie, localize at the maximal ideal $\langle x, y, x/y,x/y^2, \ldots \rangle$.  This has value group $\mathbb{Z} \oplus \mathbb{Z}$ with lexicographic ordering (in other words, the $x$-value always is more important than the $y$-value).  
It's easy to see it's not Noetherian but it does have finite Krull dimension, equal to $2$.  
You can obtain this example geometrically, and explicitly, by repeated blowings up of the origin.  See Hartshorne Chapter II, Exercise 4.12.  
