## Almost Prüfer $v$-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 $D$.