On S-valuation domains

Abstract

Let [Formula: see text] be an integral domain and let [Formula: see text] be a multiplicative subset of [Formula: see text]. In this paper, we study integral domains whose quotient rings are valuation domain. To do this, we introduce the concept of [Formula: see text]-valuation domains. We define [Formula: see text] to be an [Formula: see text]-valuation domain if for each nonzero [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text] divides [Formula: see text] or [Formula: see text] divides [Formula: see text]. Among other things, we show that [Formula: see text] is an [Formula: see text]-valuation domain if and only if [Formula: see text] is a valuation domain. By using this result, we give several valuation-like properties.