Integral domains with almost integral proper overrings

Abstract

Let R be an integral domain with quotient field K. By a proper nontrivial overring of R is meant a ring contained properly between R and K. It is proved that each proper nontrivial overring of R is almost integral over R if and only if either R is pointwise non-Archimedean (also known as anti-Archimedean) or R is a G-domain such that the complete integral closure of R is the unique one-dimensional valuation overring of R. This condition is also characterized in the class of seminormal domains by ªR is either pointwise non-Archimedean or a maximized domainº; in the class of conducive domains (where the ªmaximizedº part of the assertion can be replaced by ªR is a G-domainº); in the class of Prüfer domains (where the ªmaximizedº part of the assertion can be replaced by ªSpec R is pinched at a height 1 primeº); and in the classes of Archimedean and of Noetherian domains. In particular, each proper nontrivial overring of a valuation domain R is almost integral over R. On the other hand, if R is either Noetherian or one-dimensional, then no proper nontrivial overring of R can be both a ring of fractions of R and an almost integral extension of R.1. Introduction. All rings considered below are (commutative integral) domains. Let R be a domain, with complete integral closure R Ã and quotient field K. A key motivation for this work is the observation that if R is a conducive domain (in the sense of [7]) which is not a field, then: R has no height 1 prime ideal D R is not a G-domain D R Ã K D R is pointwise non-Archimedean (in the sense of [6], also termed anti-Archimedean in [1]). In this introduction, we sketch how the preceding observation relates to the overall plan of this paper. As the need arises in Section 2, we recall most of the definitions and basic facts concerning the above-mentioned concepts.Since each valuation domain is a conducive domain [7, Proposition 2.1], we first seek an analogue of the above observation which would be valid for arbitrary valuation domains. It is useful to say that a proper nontrivial overring of R is any ring contained properly between R and K. The desired analogue is given in Corollary 2.7 (a): if R is a valuation domain, then each (element of each) proper nontrivial overring of R is almost integral over R. Corollary 2.6 establishes that the same conclusion holds more generally, with ªvaluation domainº replaced by ªconducive divided domainº (with ªdivided domainº taken in the sense of [5]). Underlying this work is the following observation in Theorem 2.3: each proper nontrivial overring of R is almost integral over R if and only if either R is pointwise non-Archimedean or R is a G-domain such that R Ã is the only one-dimensional valuation overring of R. (In the latter case, R is a ªmaximized domainº, in the sense of [7].) This condition is interpreted