 Solution

For every valuation ring $D$ with fraction field $K$, consider compatible morphisms of schemesAn extension is a morphism of schemes $\ell :\mathrm{Spec}D\to X$ satisfying $\ell {|}_{\mathrm{Spec}K}=u$ and $f\ell =v$. The valuative criterion for universal closedness statesEquivalently, among quasi-compact morphisms, the existence condition characterizes universal closedness. A universally closed morphism is automatically a quasi-compact morphism; the converse direction of the criterion needs this compactness condition. All valuation rings are allowed, with arbitrary fraction fields; there is no Noetherian ring hypothesis and one cannot replace them by discrete valuation rings without additional assumptions.^[ref][ref]