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.

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.

By considering the irreducible components one can get the following generalisation of the above:

Let $R$ be a local reduced ring with only finitely many minimal primes whose maximal ideal $\mathfrak{m}$ is principal. Then, $R$ is noetherian if and only if its $\mathfrak{m}$-adic topology is separated.