TITLE: Examples of Non-Noetherian Valuation Rings
QUESTION [11 upvotes]: For valuation rings I know examples which are Noetherian. 

I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? 

I am very eager to know. Thanks.

REPLY [2 votes]: Let $(K, \lvert\cdot\rvert)$ be a complete algebraically closed field with a non trivial absolute value. Let $R$ be its valuation ring and $\mathfrak{m}$ the maximal ideal of $R$. Since every element of $K$ has a square root in $K$, therefore $\mathfrak{m}=\mathfrak{m}^2$. By Nakayama, $R$ cannot be noetherian. Such fields $K$ exist of course. Start with any $K$ with a non-trivial absolute value. Complete it, take the algebraic closure of the completion, and complete that. So for example the valuation ring of $\mathbb{C}_p$, with $p$ a prime number, would be such an example.