Sequence domains and integer-valued polynomials

Abstract

A problem of recent interest has been to characterize all commutative integral domains D such that Int( D) (the integer-valued polynomial ring on D) is Prufer. It is known that if D is Noetherian, then Int( D) is Prufer if and only if D is Dedekind with all residue fields finite. Moreover, it is known that if Int( D) is Prufer ( D Noetherian or not), then D is almost Dedekind with all residue fields finite. The case where D is non-Noetherian has been attacked by Chabert [2, 3], Glimer [5], and Loper [10], but is far from settled. This paper considers a special class of non-Noetherian almost Dedekind domains with finite residue fields which can be constructed by intersecting a sequence of Noetherian valuation domains which has a particular convergence property. These domains are called sequence domains. The especially simple ideal structure of sequence domains allows us to draw conclusions about the ideal structure of the integer-valued polynomial rings. For example, we show that a two-part boundedness condition proposed by Chabert in [3] completely characterizes the sequence domains D for which Int( D) is Prufer. Also, in [3] Chabert posed a condition he called “behaving well under localization” which he proved to be a sufficient condition for Int( D) to be Prufer, but left unsettled the question of its necessity. We characterize the sequence domains D for which Int( D) behaves well under localization and show by means of an example that this condition is not necessary. We construct many other examples as well, all of which are overrings of Z[ x].