Almost Prüfer -multiplication domains and the ring D+XD_S[X]
Volume 121 / 2010
Abstract
This paper is a continuation of the investigation of almost Prüfer v-multiplication domains (APVMDs) begun by Li [Algebra Colloq., to appear]. We show that an integral domain D is an APVMD if and only if D is a locally APVMD and D is well behaved. We also prove that D is an APVMD if and only if the integral closure \overline {D} of D is a PVMD, D\subseteq \overline {D} is a root extension and D is t-linked under \overline {D}. We introduce the notion of an almost t-splitting set. D^{(S)} denotes the ring D+XD_S[X], where S is a multiplicatively closed subset of D. We show that the ring D^{(S)} is an APVMD if and only if D^{(S)} is well behaved, D and D_S[X] are APVMDs, and S is an almost t-splitting set in D.