Graded almost pseudo-valuation domains

Abstract

Let Γ be a nonzero commutative cancellative monoid (written additively), $R=\underset{\alpha \in \Gamma }{\oplus }{R}_{\alpha }$ be a Γ-graded integral domain with $R_{\alpha }\ne \left\{0\right\}$ for all $\alpha \in \Gamma$, and H be the set of nonzero homogeneous elements of R. A homogeneous ideal P of R will be said to be strongly homogeneous primary if $xy\in P$ implies $x\in P$ or $y^n\in P$ for some integer $n\ge 1$, for every homogeneous elements x, y of R_H. We say that R is a graded almost pseudo-valuation domain (gr-APVD) if each homogeneous prime ideal of R is strongly homogeneous primary. In this paper, we study some ring-theoretic properties of gr-APVDs and graded integral domains R such that $R_{H\setminus P}$ is a gr-APVD for all homogeneous maximal ideals (resp., homogeneous maximal t-ideals) P of R.

Keywords:

• Graded almost pseudo-valuation domain
• graded integral domain
• pseudo-valuation domain
• UMt-domain

2020 Mathematics Subject Classification:

• 13A02
• 13A15
• 13F05

Acknowledgments

The authors would like to thank Gyu Whan Chang and Nematollah Shirmohammadi, for their comments on an earlier version of this paper.

Additional information

Funding

Haleh Hamdi disclosed receipt of the following financial support for the research: This research was in part supported by a grant from IPM (No. 1402130024).