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| Paper IPM / M / 17599 |
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| Abstract: | |
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Let $\Gamma$ be a nonzero commutative cancellative monoid (written additively), $R = \bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain with $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, and $H$ the saturated multiplicative set of nonzero homogeneous elements of $R$. A homogeneous prime ideal $P$ of $R$ is said to be a pseudo strongly homogeneous prime ideal if for each homogeneous elements $x, y\in R_H$ whenever $xyP\subseteq P$, then there exists a positive integer $n$, such that either $x^n \in R$ or $y^nP \subseteq P$.
A graded integral domain $R$ is said to be a graded pseudo-almost valuation domain (gr-PAVD) if each homogeneous prime ideal of $R$ is a pseudo-strongly homogeneous prime ideal. We study the prime ideal- and ring-theoretic properties and overrings of gr-PAVDs. We also study the gr-PAVD property in pullback of graded domains and give various examples of these domains.
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