Noetherianity of valuation ring and valuation being discrete

Question

I need a hint for left to right part of the following:

Let $K$ be a valued field with $\nu$ and $\mathcal{O}_\nu$ be its valuation ring. Then, $\mathcal{O}_\nu$ is Noetherian if and only if $\nu$ is discrete.

My intuition is, considering the contrapositive, if $\nu$ is not discrete, then there will be a non-integer valuation $q$ and the ideals (corresponding valuations) "converging" to $q$ would not satisfy the ascending chain condition.

Is it more or less a right track? In case not, I would appreciate any help...

Answer 1

Hint: Recall that in a valuation ring, the ideals are linearly ordered.

So, if $\mathcal{O}_v$ is Noetherian, then $\mathfrak{m}_v=(a_1,\ldots,a_n)$, and by the above we may assume that $(a_1)\subseteq\cdots\subseteq (a_n)$ so...