Prufer v-multiplication domains and related domains of the form D + D-S[Gamma*]

Abstract

Let D be an integral domain. S be a saturated multiplicative subset of D with D subset of D-S, and Gamma be a nonzero torsion-free grading monoid with Gamma boolean AND - Gamma = {0}. Let D-S[Gamma] be the semigroup ring of Gamma over D-S, Gamma* = Gamma - {0}, and D-(S,D-Gamma) = D + D-S[Gamma*], i.e., D-(S,D-Gamma) = {f is an element of D-S[Gamma] vertical bar f(0) is an element of D}. We show that D-(S,D-Gamma) is a PvMD (resp., GCD-domain, GGCD-domain) if and only if D is a PvMD (resp., GCD-domain, GGCD-domain), Gamma is a valuation semigroup and S is a t-splitting (resp., splitting, d-splitting) set of D. (C) 2010 Elsevier Inc. All rights reserved.