Integrally closed domains with treed overrings

On Seminormal Integral Domains With Treed Overrings

“…(Thus, although our applications are to domains, we deviate from some of the literature on the FCP property where "ring" has tacitly meant "domain".) Theorem 2.5 leads directly to Corollary 2.6, where we give our extension of the motivating result of Ayache [3,Theorem 10]. Given the role of pseudo-valuation domains in [3,Theorem 10] and Corollary 2.6, it seems worthwhile to deepen the study of pseudo-valuation domains; this work does so through the inclusion of two results (Propositions 2.1 and 2.9) from the unpublished dissertation of M. S. Gilbert [26].…”

“…Theorem 2.5 leads directly to Corollary 2.6, where we give our extension of the motivating result of Ayache [3,Theorem 10]. Given the role of pseudo-valuation domains in [3,Theorem 10] and Corollary 2.6, it seems worthwhile to deepen the study of pseudo-valuation domains; this work does so through the inclusion of two results (Propositions 2.1 and 2.9) from the unpublished dissertation of M. S. Gilbert [26]. These two results are not needed for Theorem 2.5 or its applications, but as explained below, they do provide some motivation for the formulation of Corollary 2.6.…”

“…Subsequently, the investigation was deepened by Ayache, Jarboui and Massaoud [5], in a study of pairs of domains all of whose intermediate rings are treed. Very recently, Ayache [3,Theorem 10] has proved that if an integrally closed domain R is such that each overring of R is a treed domain, then R is an LPVD. This raises the following question, which returns to the spirit of [12]: what can be said along the lines of [3] if R is not integrally closed (but is such that each of its overrings is treed)?…”

Characterizations of the Integral Domains Whose Overrings Are Going-Down Domains

“…In fact, by an iterated pullback construction, it was shown in [19] that an integral overring of a going-down domain need not be a going-down domain. (Earlier, it had been shown that each integral overring of a going-down domain is a going-down domain if dim v (R) ≤ 2 [14] or if R is both locally divided and locally finite-conductor [ [3,Theorem 10]. An underlying difficulty in such studies is that the nature of R has very subtle influences on its overrings.…”

“…An underlying difficulty in such studies is that the nature of R has very subtle influences on its overrings. In addressing the general case, we have found inspiration from two sources: Ayache's recent result [3] that if an integrally closed domain R is such that each overring of R is treed, then R must be an LPVD; and the role, in the above-cited result from [12] If R is an LPVD and M is a maximal ideal of R, it will be convenient to let V R (M ) denote the valuation domain which is canonically associated to the pseudovaluation domain R M . Also, ⊂ will denote proper inclusion.…”