Characterizations of the integral domains whose overrings are going-down domains (Q2879414)

Throughout, let \(R\) be an integral domain with quotient field \(K\). By an overring of \(R\) we shall mean a ring \(T\) between \(R\) and \(K.\) \(R\) is said to be a going down (GD-) domain if for every overring \(T\) of \(R\) the extension \( R\subseteq T\) satisfies the GD-property. Next \(R\) is a pseudo-valuation domain (PVD) if every prime ideal \(P\) of \(R\) is strongly prime, i.e., for all \(x,y\in K\) \(xy\in P\) implies that \(x\in P\) or \(y\in P\) [\textit{J. R. Hedstrom} and \textit{E. G. Houston}, Pac. J. Math. 75, 137--147 (1978; Zbl 0368.13002)]. \textit{J. R. Hedstrom} and \textit{E. G. Houston} [Houston J. Math. 4, 199--207 (1978; Zbl 0416.13014)] also showed that \(R\) is a PVD if and only if there is a uniquely determined valuation ring \(V\) such that \(\mathrm{Spec}(R) = \mathrm{Spec}(V)\), as sets; this \(V\) is called the canonically associated valuation ring of the PVD \(R.\) The second author and \textit{M. Fontana} [Ann. Mat. Pura Appl. (4) 134, 147--168 (1983; Zbl 0531.13012)] called \(R\) locally PVD (LPVD) if \(R_{M}\) is a PVD for each maximal ideal \(M\) of \(R\). The authors use \(\dim _{v}(R)\) to denote the valuative dimension of \(R\) and it is best described as: \(\dim _{v}(R)=\sup\{\dim(V):V\) valuation overring of \(R\}\). \(R\) is called a Jaffard domain if \(\dim _{v}(R)=\dim (R)<\infty .\) Finally, \(R\) is catenarian if for each pair of prime ideals \(P\subseteq Q\) all saturated chains of prime ideals going from \(P\) to \(Q\) have the same finite length.NEWLINENEWLINEIt was shown by the second author in [Commun. Algebra 1, 439--458 (1974; Zbl 0285.13001)] that a GD-domain \(R\) must be treed, i.e., \(\mathrm{Spec}(R)\), the set of prime ideals of \(R,\) is a tree under inclusion. However \(\mathrm{Spec}(R)\) being a tree does not mean that \(R\) must be a GD-domain; this result was established by Lewis as indicated by the second author and \textit{I. J. Papick} in Example 2.2 of [Nieuw Arch. Wiskd., III. Ser. 26, 255--291 (1978; Zbl 0383.13005)]. Next, an overring of a GD-domain need not be a GD-domain, as shown by the second author and \textit{M. Fontana} in [Proc. Am. Math. Soc. 115, No. 3, 655--662 (1992; Zbl 0759.13004)]. The main goal of the paper under review is to characterize domains each of whose overrings is treed. This oft attempted problem saw the ray of light with the following result of the first author: If each overring of an integrally closed domain \(R\) is treed then \(R\) must be an LPVD [\textit{A. Ayache}, Ric. Mat. 63, No. 1, 93--100 (2014; Zbl 1301.13007)].NEWLINENEWLINE The main theorem of the paper and its consequences are best described by the authors themselves:NEWLINENEWLINE ``Our main result, Theorem 2.8, provides the following characterization for the general case: each overring of (a domain) \(R\) is a going-down domain if and only if \( R^{\prime }\) is an LPVD, \(T\subseteq T^{\prime }\) satisfies going-down for every overring \(T\) of \(R\), and \(\mathrm{tr.deg}[V_{R^{\prime }}(M)/M: R^{\prime }/M]\) \(\leq 1\) for every maximal ideal \(M\) of \(R^{\prime }\) (where \(V_{R^{\prime }}\) \((M)\) denotes the valuation domain that is canonically associated to the pseudo-valuation domain \((R^{\prime })_{M}).\) Theorem 2.8 also shows that in case \(R\) is locally finite-dimensional, the above conditions are equivalent to: each overring \(T\) of \(R\) is treed and satisfies \(\mathrm{ht}_{T\prime }\) \((Q)\) =\(\mathrm{ht}_{T}\) \((Q\cap R)\) for every \(Q\in \mathrm{Max}(T^{\prime }).\) Some other interesting applications are also given along these lines, including the case where \(R\) is integrally closed (Proposition 2.2) or \(R\) is not a Jaffard domain (Proposition 2.5) or \(R[X]\) is catenarian (Corollary 2.11).''