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

### Tom 121 / 2010

Colloquium Mathematicum 121 (2010), 239-247 MSC: 13G05, 13A15, 13F20. DOI: 10.4064/cm121-2-6

#### Streszczenie

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 $D$.

#### Autorzy

• Qing LiCollege of Computer Science and Technology
Southwest University for Nationalities
Chengdu 610041, P.R. China
e-mail

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