Example of non finitely-generated $R$-Module $K$ so that $K/PK$ is finitely generated

Question

I am studying about this and questioning some problems.

Suppose $F$ is a field and $R=F[x]_{(x)}$, the localisation of $F[x]$ at prime ideal $P=(x)$. I am trying to find a non finitely-generated $R$-module $K$, but making $K/PK$ is finitely generated, but still could not. Can anyone give me the example and the explanation please.

Thank you.

I am a newbie here, hope we can help each other in studying Math :D

Answer 1

How about taking $K$ equal to the field of fractions of $R$?