Let $R$ be a local domain whose maximal ideal $\mathfrak{m}$ is principal. Then, $R$ is noetherian if and only if its $\mathfrak{m}$-adic topology is separated. If $R$ is moreover not a field, then it is noetherian if and only if it is a discrete valuation ring. For proofs see Bourbaki's _Algèbre commutative,_ VI.4.6 Proposition 9.