Let $R$ be an integral domain such that the localization, $R_{\mathfrak p}$, at each prime ideal, $\mathfrak p \le R$ is Noetherian. Then is $R$ necessarily Noetherian?
In the case of $R$ not neccesarily being a domain, I know of a couple of examples to show that $R$ need not being Noetherian (e.g. this & this). However, in each example, $R$ is not a domain.