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It is well-known that an integrally closed domain $D$ can be express as theintersection of its valuation overrings but, if $D$ is not a Pr\"{u}fer domain,the most of valuation overrings of $D$ cannot be seen as localizations of $D$.The Kronecker function ring of $D$ is a classical construction of a Pr\"{u}ferdomain which is an overring of $D[t]$, and its localizations at prime idealsare of the form $V(t)$ where $V$ runs through the valuation overrings of $D$.This fact can be generalized to arbitrary integral domains by expressing themas intersections of overrings which admit a unique minimal overring. In thisarticle we first continue the study of rings admitting a unique minimaloverring extending known results obtained in the 70's and constructing exampleswhere the integral closure is very far from being a valuation domain. Then weextend the definition of Kronecker function ring to the non-integrally closedsetting by studying intersections of Nagata rings of the form $A(t)$ for $A$ anintegral domain admitting a unique minimal overring.

Non-integrally closed Kronecker function rings and integral domains with a unique minimal overring