On Going Down for Simple Overrings

Abstract

Abstract.Let R be an integral domain with quotient field K. If R is Noetherian : then the Krull dimension of R is at most l<=>for all overrings 5 of R, R<=S satisfies going down. R is Dedekind (resp., PID)oÄ is Krull (resp., UFD) and, for all uG K, R<=R [u] satisfies going down. R is Priifero/?is integrally closed, every intersection of two principal ideals of R is finitely generated, and R^R [u] satisfies going down for all uE K.

Integral Overrings of Two-Dimensional Going-Down Domains

“…Let R be a (commutative integral) domain. As in [6,12], we say that R is a going-down domain if R c T satisfies GD (in the sense of [17, p. 28]) for each domain T containing R . Natural examples of going-down domains include Prüfer domains, domains of (Krull) dimension 1, and certain D + M constructions (cf.…”

“…Natural examples of going-down domains include Prüfer domains, domains of (Krull) dimension 1, and certain D + M constructions (cf. [12,Corollary]). In some respect, the class W of going-down domains is well-behaved; for instance, being a going-down domain is a local property (cf.…”

“…In some respect, the class W of going-down domains is well-behaved; for instance, being a going-down domain is a local property (cf. [6,Lemma 2.1]). Moreover, certain subclasses of & are known to be stable under integral extensions (cf.…”

“…Indeed, by analyzing a construction of Heinzer-Ohm [16, p. 6], Dobbs showed in [7,Example 2.1] that an integral extension of a two-dimensional valuation domain need not be a going-down domain. By applying the classical D + M construction and [ 12,Corollary], we see for each « , 3 < « < oo, that there exists an «-dimensional Ä e ? such that R has an integral overling which is not in W. (It follows from [8,Lemma 2.2(b)] that the domains R in these examples are, in fact, divided domains, an important type of quasilocal going-down domain studied in [8].)…”

Going-down underrings

“…Beginning with [77], a number of papers have considered this question in the case where "small" means "Noetherian." Here, we shall interpret "small" as meaning variously "of (Krull) dimension at most 1," "a goingdown domain" (in the sense of [3]), or "treed." Of course, these three interpretations of "small" need not be equivalent for a given ring.…”