Let $X$ be an integral noetherian scheme, let $\xi$ be its generic point. Then it is not so hard to show that $\{ \xi\}$ is open in $X$ if and only if $X$ is a finite set. In termes of algebra, it says the following:
Let $A$ be a noetherian integral domain. Suppose there exists $f\in A$ non-zero such that the localization $A_f$ is a field. Then $A$ is semi-local of Krull dimension at most $1$.
Question: What is the name of this result ? I am sure I saw it once somewhere.
The motivation comes from a construction of non-discrete zero-dimensional schemes here.
