Commutative rings with invertible-radical factorization

TL;DR

This paper explores a special class of rings where ideals can be factored into invertible and radical ideals, examining their stability under various constructions and introducing new examples of such rings.

Contribution

It introduces and studies rings with invertible-radical ideal factorizations, analyzing their stability under homomorphic images and various ring constructions, and provides new examples.

Findings

Identified conditions for stability under homomorphic images.

Established transfer properties to direct products, extensions, and duplications.

Generated new families of rings with invertible-radical factorizations.

Abstract

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.