Almost Prüfer $v$-multiplication domains and the ring $D+XD_S[X]$

Qing Li¹

¹ College of Computer Science and Technology Southwest University for Nationalities Chengdu 610041, P.R. China

Colloquium Mathematicum, Tome 121 (2010) no. 2, pp. 239-247

Résumé

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 $D$.

Détail

DOI : 10.4064/cm121-2-6

Mots-clés : paper continuation investigation almost fer v multiplication domains apvmds begun algebra colloq appear integral domain apvmd only locally apvmd behaved prove apvmd only integral closure overline pvmd subseteq overline root extension t linked under overline introduce notion almost t splitting set denotes ring where multiplicatively closed subset ring apvmd only behaved apvmds almost t splitting set