Graded primitive and INC-extensions
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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.
Abstract
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.
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