Commutative rings in which every finitely generated ideal is quasi-projective

TL;DR

This paper explores the structure of commutative rings where every finitely generated ideal is quasi-projective, positioning these rings between arithmetical and Gaussian rings, and extends known theorems and conjectures in this area.

Contribution

It introduces a new class of rings with quasi-projective finitely generated ideals, generalizes existing theorems, and provides novel examples through trivial ring extensions.

Findings

The class lies strictly between arithmetical and Gaussian rings.

Generalizes Osofsky's theorem on weak global dimension.

Partially resolves Bazzoni-Glaz's conjecture on Gaussian rings.

Abstract

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prufer conditions between a ring and its total ring of quotients. Section 4 examines various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prufer rings.