A characterization of graded pseudo-valuation domains

Abstract

Let [Formula: see text] be a torsionless grading monoid, [Formula: see text] a [Formula: see text]-graded integral domain, [Formula: see text] the set of nonzero homogeneous elements of [Formula: see text], [Formula: see text] the quotient field of [Formula: see text] and [Formula: see text] the group of units of [Formula: see text]. We say that [Formula: see text] is a graded pseudo-valuation domain (gr-PVD) if whenever a homogeneous prime ideal [Formula: see text] of [Formula: see text] contains the product [Formula: see text] of two homogeneous elements of [Formula: see text], then [Formula: see text] or [Formula: see text]. 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 [Formula: see text] is a gr-PVD if and only if exactly one of the following two conditions holds: (1) (a) [Formula: see text], (b) [Formula: see text] is a pseudo-valuation monoid, and (c) [Formula: see text] for every [Formula: see text] whenever [Formula: see text] is not a unit. (2) (a) [Formula: see text], (b) [Formula: see text] is a valuation monoid, (c) [Formula: see text] for every [Formula: see text] whenever [Formula: see text] is not a unit, and (d) [Formula: see text] is a gr-PVD.