An integral domainRis said to be a UMT-domain if uppers to zero inR[X) are maximalt-ideals. We show thatRis a UMT-domain if and only if its localizations at maximaltdeals have Prüfer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prüfer integral closure if and only if all its overrings are UMT-domains.