Journal of the
Korean Mathematical Society JKMS

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.

Keywords: Graded pseudo-valuation ring, graded trivial ring extension, graded amalgamated algebra along an ideal

MSC numbers: 13A02, 13E05

Supported by: The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2021R1I1A3047469).