On a subclass of semistar going-down domains

چکیده مقاله

Let $D$ be an integral domain and let $\star$ be a semistar
operation on $D$. In this paper, we define the class of
$\star$-quasi-going-up domains, a notion dual to the class of
$\star$-going-down domains. It is shown that the class of
$\star$-quasi-going-up domains is a proper subclass of
$\star$-going-down domains and that every
Pr\"{u}fer-$\star$-multiplication domain is a $\star$-quasi-going-up
domain. Next, we prove that the $\star$-Nagata ring $\Na(D,\star)$,
is a quasi-going-up domain if and only if $D$ is a
$\widetilde{\star}$-quasi-going-up and a
$\widetilde{\star}$-quasi-Pr\"{u}fer domain. Several new
characterizations are given for $\star$-going-down domains. We also
define the universally $\star$-going-down domains, and then, give
new characterizations of Pr\"{u}fer-$\star$-multiplication domains.