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Let Γ be a torsionless grading monoid and R = ⊕ α ∈ Γ R α be a Γ -graded integral domain. In this paper, we present basic properties and give a complete characterization of strongly homogeneous prime ideals of R. These works generalize Ahmed and the authors' characterization of when R is a graded pseudo-valuation domain. [ABSTRACT FROM AUTHOR]

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]

In this paper, we introduce and study a generalization of powerful ideals, in the sense of Badawi and Houston, to the context of integral domains graded by a torsionless monoid. [ABSTRACT FROM AUTHOR]