Pairs of integral domains with most of the intermediate rings PVD
A PVD is a pseudo-valuation domain, i.e. a ring \(R\) such that there is a valuation ring \(V\) with \(R\subset V\subset \text{Frac}(R)\) and Spec\((R)=\) Spec\((V)\). The main result of the paper under review is that, given an extension of integral domains \(R\subset S\), if every proper subring of \(S\) containing \(R\) is a PVD then \(S\) is a PVD. From this the authors deduce that, given an extension of integral domains \(R\subset S\), if every proper subring of \(S\) containing \(R\) is a valuation domain (resp. a DVR) then \(S\) is a also a valuation domain (resp. the quotient field of \(R\)).
scientific article; zbMATH DE number 6815066
scientific article