We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals

Abstract. We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are the atomic pseudo-valuation domains.

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.

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In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring along an ideal. Our results generate examples that enrich the current literature with new and original families of rings satisfying these properties.