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Commutative rings with invertible-radical factorization. (English) Zbl 1498.13018

Recall that an ISP-domain (after Ahmed-Dumitrescu) is an integral domain whose ideals can be factored as an invertible ideal times a nonempty product of proper radical ideals. It is known that an ISP-domain is a strongly discrete Prüfer (i.e., a Prüfer domain having no idempotent prime ideal except the zero ideal) and every nonzero prime ideal is contained in a unique maximal ideal.
In the present paper, the authors extend the ISP-domain concept to rings with zero-divisors in two different ways. The first class of rings consists of those rings in which every proper regular ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals (called ISP-rings). The second class consists of those rings in which every proper ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals (called strongly ISP-rings).
This paper is devoted to the investigation of 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. As applications of their main results, the authors produce several classes of interesting examples that enrich the literature with new and original families of rings satisfying the above mentioned factorisation properties.

MSC:

13B99 Commutative ring extensions and related topics
13A15 Ideals and multiplicative ideal theory in commutative rings
13G05 Integral domains
13B21 Integral dependence in commutative rings; going up, going down
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References:

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