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Is there a non-tautological characterization of the class of commutative unital Noetherian rings satisfying the following property: for any open subset $U$ of the topological space of $\mathrm{Spec}\,R$, the natural map from the localization of $R$ at the multiplicative set of global functions not vanishing anywhere on $U$ to $\mathcal{O}_{\mathrm{Spec}\,R}(U)$ is an isomorphism? Some interesting sufficient conditions at least?

Note that not all commutative unital Noetherian rings satisfy this property.

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