Weak global dimension of Prufer-like rings

TL;DR

This paper surveys recent research on the weak global dimension of Prufer-like rings, highlighting key results, conjectures, and examples, and aims to make the topic accessible to a broad audience.

Contribution

It provides a comprehensive overview of recent developments and open problems regarding the weak global dimension of Prufer-like rings, including detailed explanations and illustrative examples.

Findings

Osofsky's proof of infinite projective dimension modules on chained rings

Validation of the Bazzoni-Glaz conjecture for fqp-rings

Existence of Gaussian rings with infinite weak global dimension that are neither arithmetical nor coherent

Abstract

In 1969, Osofsky proved that a chained ring (i.e., local arithmetical ring) with zero divisors has infinite weak global dimension; that is, the weak global dimension of an arithmetical ring is 0, 1, or infinite. In 2007, Bazzoni and Glaz studied the homological aspects of Prufer-like rings, with a focus on Gaussian rings. They proved that Osofsky's aforementioned result is valid in the context of coherent Gaussian rings (and, more generally, in coherent Prufer rings). They closed their paper with a conjecture sustaining that ``the weak global dimension of a Gaussian ring is 0, 1, or infinite. In 2010, Bakkari-Kabbaj-Mahdou provided an example of a Gaussian ring which is neither arithmetical nor coherent and has an infinite weak global dimension. In 2011, Abuhlail-Jarrar-Kabbaj introduced and investigated the new class of fqp-rings which stands strictly between the two classes of arithmetical rings and Gaussian rings. Then, they proved the Bazzoni-Glaz conjecture for fqp-rings. This paper surveys a few recent works in the literature on the weak global dimension of Prufer-like rings making this topic accessible and appealing to a broad audience. As a prelude to this, the first section of this paper provides full details for Osofsky's proof of the existence of a module with infinite projective dimension on a chained ring. Numerous examples -arising as trivial ring extensions- are provided to illustrate the concepts and results involved in this paper.