Integral domains that lose ideals in overrings (Q1106273)

We study domains with the property that, in any proper overring, some proper ideal extends to the unit ideal, and domains with the property that, in any proper overring, some proper ideal of a certain type (v- ideal, invertible ideal, or principal ideal) extends to the unit ideal. We characterize the Noetherian domains with these properties, and we show that even the strongest of these properties does not imply the QR- property.