Communications of the
Korean Mathematical Society CKMS

It is well-known that quasi-Pr\"{u}fer domains are characterized as those domains $D$, such that every extension of $D$ inside its quotient field is a primitive extension and that primitive extensions are characterized in terms of INC-extensions. Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain graded by an arbitrary torsionless grading monoid $\Gamma$ and $\star$ be a semistar operation on $R$. The main purpose of this paper is to give new characterizations of gr-$\star$-quasi-Pr\"{u}fer domains in terms of graded primitive and INC-extensions. Applications include new characterizations of UM$t$-domains.

Keywords: semistar operation, graded domain, UM$t$-domain, primitive extension, INC-extension

MSC numbers: Primary 13A15, 13G05, 13A02, 13F05