On semistar Nagata rings, Prufer-like domains and semistar going-down domains

چکیده مقاله

Let $\star$ be a semistar operation on a domain $D$. Then the semistar Nagata ring $\Na(D, \star)$ is a treed domain $\Leftrightarrow \, D$ is $\widetilde{\star}$-treed and the contraction map $\Spec(\Na(D,\star))\to
\QSpec^{\widetilde{\star}}(D)\cup\{0\}$ is a bijection $\Leftrightarrow \, D$ is a $\widetilde{\star}$-treed and $\widetilde{\star}$-quasi-Pr\"ufer domain. Consequently, if $D$ is a $\widetilde{\star}$-Noetherian domain but not a field, then
$D$ is $\widetilde{\star}$-treed if and only if $\widetilde{\star}$-$\dim(D)=1$. The ring $\Na(D, \star)$ is a
going-down domain if and only if $D$ is a $\widetilde{\star}$-$\GD$ domain and a $\widetilde{\star}$-quasi-Pr\"ufer domain. In general, $D$ is a P$\star$MD $\Leftrightarrow \, \Na(D,\star)$ is an integrally closed treed domain $\Leftrightarrow \, \Na(D,\star)$ is an integrally closed going-down domain. If $P$ is a quasi-$\star$-prime ideal of $D$, an induced stable semistar operation of finite type, $\star/P$, is defined on $D/P$. The associated Nagata rings satisfy
$\Na(D/P,\star/P)\cong \Na(D,\star)/P\Na(D,\star)$. If $D$ is a P$\star$MD (resp., a $\widetilde{\star}$-Noetherian domain; resp., a $\star$-Dedekind domain; resp., a $\widetilde{\star}$-$GD$ domain), then $D/P$ is a P($\star/P$)MD (resp., a $(\star/P)$-Noetherian domain; resp., a $(\star/P)$-Dedekind domain; resp., a $(\star/P)$-$GD$ domain).