Arithmetic properties in pullbacks (Q876349)

Let \(I\) be a nonzero ideal of a (commutative) domain \(T\), and let \(R\) be a proper subring of \(T\) containing \(I\) so that \(I\) is a prime ideal of \(R\). Thus \(R\) can be described as a pullback. The authors present characterizations for various properties of \(R\): Prüfer, Bézout, and with some additional assumptions, also for the PVMD, \(v\)-domain and GCD-domain properties. For example, \(R\) is a Prüfer domain iff \(D:=R/I\) and \(T\) are Prüfer domains, \(I\) is a prime ideal of \(T\), and the domains \(D\) and \(T/I\) have the same quotient fields. The authors introduce the \(\tilde v\) and the \(\tilde t\) operations for domains \(D\subseteq E\) with the same quotient field; using these operations they define \(E\)-PVMDs, \(E\)-\(v\)-domains and \(E\)-GCD-domains. They use these properties to obtain some of their characterizations.