On almost valuation ring pairs

Abstract

If [Formula: see text] are (commutative) rings, [Formula: see text] denotes the set of intermediate rings and [Formula: see text] is called an almost valuation (AV)-ring pair if each element of [Formula: see text] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let [Formula: see text] be rings, with [Formula: see text] denoting the integral closure of [Formula: see text] in [Formula: see text]. Then [Formula: see text] is an AV-ring pair if and only if both [Formula: see text] and [Formula: see text] are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions [Formula: see text]. If [Formula: see text] is an AV-ring pair, then [Formula: see text] is a P-extension. The AV-ring pairs [Formula: see text] arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of [Formula: see text] type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for [Formula: see text] being an AV-ring pair to entail that [Formula: see text] is an overring of [Formula: see text], but there exist domain-theoretic counter-examples to such a conclusion in general. If [Formula: see text] is an AV-ring pair and [Formula: see text] satisfies FCP, then each intermediate ring either contains or is contained in [Formula: see text]. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.