A characterization of graded pseudo-valuation domains.

Let Γ be a torsionless grading monoid, R = ⊕ α ∈ Γ R α a Γ -graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R 0 and G 0 the group of units of Γ. We say that R is a graded pseudo-valuation domain (gr-PVD) if whenever a homogeneous prime ideal P of R contains the product xy of two homogeneous elements of R H , then x ∈ P or y ∈ P. The notion of gr-PVDs was introduced recently by the authors in (M. T. Ahmed, C. Bakkari, N. Mahdou and A. Riffi, Graded pseudo-valuation domains, Comm. Algebra 48 (2020) 4555–4568) as a graded version of pseudo-valuation domains (PVDs). In this paper, we show that R is a gr-PVD if and only if exactly one of the following two conditions holds : (1)(a) R 0 = K , (b) Γ is a pseudo-valuation monoid, and (c) R α = (R H) 0 x for every 0 ≠ x ∈ R α whenever α ∈ Γ is not a unit. (2)(a) R 0 ≠ K , (b) Γ is a valuation monoid, (c) R α = K x for every 0 ≠ x ∈ R α whenever α ∈ Γ is not a unit, and (d) T = ⊕ α ∈ G 0 R α is a gr-PVD. [ABSTRACT FROM AUTHOR]