Uppers to zero in polynomial rings over graded domains and UM$t$-domains
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Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain, $H$ be the set of nonzero homogeneous elements of $R$, and $\star$ be a semistar operation on $R$. The purpose of this paper is to study the properties of quasi-Pr\"{u}fer and UM$t$-domains of graded integral domains. For this reason we study the graded analogue of $\star$-quasi-Pr\"{u}fer domains called gr-$\star$-quasi-Pr\"{u}fer domains. We study several ring-theoretic properties of gr-$\star$-quasi-Pr\"{u}fer domains. As an application we give new characterizations of UM$t$-domains. In particular it is shown that $R$ is a gr-$t$-quasi-Pr\"{u}fer domain if and only if $R$ is a UM$t$-domain if and only if $R_P$ is a quasi-Pr\"{u}fer domain for each homogeneous maximal $t$-ideal $P$ of $R$. We also show that $R$ is a UM$t$-domain if and only if $H$ is a $t$-splitting set in $R[X]$ if and only if each prime $t$-ideal $Q$ in $R[X]$ such that $Q\cap H=\emptyset$ is a maximal $t$-ideal.
Abstract
Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain, $H$ be the set of nonzero homogeneous elements of $R$, and $\star$ be a semistar operation on $R$. The purpose of this paper is to study the properties of quasi-Pr\"{u}fer and UM$t$-domains of graded integral domains. For this reason we study the graded analogue of $\star$-quasi-Pr\"{u}fer domains called gr-$\star$-quasi-Pr\"{u}fer domains. We study several ring-theoretic properties of gr-$\star$-quasi-Pr\"{u}fer domains. As an application we give new characterizations of UM$t$-domains. In particular it is shown that $R$ is a gr-$t$-quasi-Pr\"{u}fer domain if and only if $R$ is a UM$t$-domain if and only if $R_P$ is a quasi-Pr\"{u}fer domain for each homogeneous maximal $t$-ideal $P$ of $R$. We also show that $R$ is a UM$t$-domain if and only if $H$ is a $t$-splitting set in $R[X]$ if and only if each prime $t$-ideal $Q$ in $R[X]$ such that $Q\cap H=\emptyset$ is a maximal $t$-ideal.
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