Let $X$ be an affine algebraic set over an algebraically closed field $k$. Let $\phi: k[X] \to R$ be a morphism of $k$-algebras. One can ask if there exists an open affine subvariety $U \subseteq X$ such that the the morphism $i^*: k[X] \hookrightarrow k[U]$ (induced by the inclusion $i: U \hookrightarrow X$) makes $k[U]$ isomorphic to $R$ as a $k[V]$-algebra. Is there a (relatively nice) algebraic condition on the map $\phi: k[X] \to R$ which determines when this is the case?
Note for example that $k[X],k[U]$ are finitely generated algebras over $k$, so $k[U]$ is always finitely generated over $k[X]$, which gives a neccesay condition on $\phi$.
Similarly, if $X$ is irreducible then we have $$k[U] = \{f/g: V(g)\cap U = \emptyset\} $$ which is the localisation of $k[X]$ at the set $S=\{g \in k[U]: V(g)\cap U = \emptyset\}$, so we know that $k[X] \to k[U]$ must be a localisation. However, it can't be any localisation because for instance $k(X) \cong \textrm{Frac}(k[X])$ is not finitely generated over $k[X]$.