Graded Pseudo-Valuation Rings
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Let $R=\bigoplus_{\alpha \in \Gamma} R_{\alpha}$ be a commutative ring graded by an arbitrary torsionless monoid $\Gamma$. A homogeneous prime ideal $P$ of $R$ is said to be strongly homogeneous prime if $aP$ and $bR$ are comparable for any homogeneous elements $a,b $ of $R$. We will say that $R$ is a graded pseudo-valuation ring (gr-PVR for short) if every homogeneous prime ideal of $R$ is strongly homogeneous prime. In this paper, we introduce and study the graded version of the pseudo-valuation rings which is a generalization of the gr-pseudo-valuation domains in the context of arbitrary $\Gamma$-graded rings (with zero-divisors). We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of $\Gamma$-graded rings that satisfy the above mentioned property.
Abstract
Let $R=\bigoplus_{\alpha \in \Gamma} R_{\alpha}$ be a commutative ring graded by an arbitrary torsionless monoid $\Gamma$. A homogeneous prime ideal $P$ of $R$ is said to be strongly homogeneous prime if $aP$ and $bR$ are comparable for any homogeneous elements $a,b $ of $R$. We will say that $R$ is a graded pseudo-valuation ring (gr-PVR for short) if every homogeneous prime ideal of $R$ is strongly homogeneous prime. In this paper, we introduce and study the graded version of the pseudo-valuation rings which is a generalization of the gr-pseudo-valuation domains in the context of arbitrary $\Gamma$-graded rings (with zero-divisors). We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of $\Gamma$-graded rings that satisfy the above mentioned property.
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