Weak Global Dimension of Prüfer-Like Rings

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 ∞. In 2007, Bazzoni and Glaz studied the homological aspects of Prüfer-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 Prüfer rings). They closed their paper with a conjecture sustaining that “the weak global dimension of a Gaussian ring is 0, 1, or ∞.” In 2010, the authors of Bakkari et al. (J. Pure Appl. Algebra 214:53–60, 2010) provided an example of a Gaussian ring which is neither arithmetical nor coherent and has an infinite weak global dimension. In 2011, the authors of Abuihlail et al. (J. Pure Appl. Algebra 215:2504–2511, 2011) 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 Prüfer-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.