On almost Prüfer \(v\)-multiplication domains (Q2909078)

Let \(R\) be an integral domain and \(R^\prime\) be the integral closure of \(R\) in its fraction field. The author introduces the notion of an almost Prüfer \(v\)-multiplication domain (APVMD) as a generalization of the concept of a Prüfer \(v\)-multiplication domain (PVMD) (Recall that \(R\) is a PVMD if (a,b) is \(t\)-invertible for any \(a,b\in R \backslash \{0\}\)). An integral domain \(R\) is said to be an APVMD if for any \(a,b\in R\backslash \{0\}\) there is \(n\in \mathbb{N}\) such that \((a^n,b^n)\) is \(t\)-invertible. The author shows that the class of APVMDs coincides with the class of \(t\)-locally almost valuation domains and that the class of PVMDs coincides with the class of integrally closed (root closed) APVMDs. An alternative characterization of APVMDs in terms of UMT-domains is also given. Namely, \(R\) is an APVMD if and only if \(R\) is an UMT-domain and \(R\subseteq R^\prime\) is a root extension. The author also characterizes an almost GCD-domain as an APVMD with torsion \(t\)-class group in \textit{D. D. Anderson} and \textit{M. Zafrullah} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 51(99), No. 1, 3--9 (2008; Zbl 1199.13023)] it has been proved that an integrally closed domain \(R\) is an almost GCD-domain if and only if \(R\) is a PVMD with torsion \(t\)-class group).NEWLINENEWLINEThe behavior of the APVMD property in the context of ring extensions is also studied (namely polynomial extensions, \(t\)-linked overings, as well as \(R+x\big(\text{Frac}(R)[x]\big)\)).