An Alternative Perspective on Projectivity of Modules

Abstract

We approach the analysis of the extent of the projectivity of modules from a fresh perspective as we introduce the notion of relative subprojectivity. A module M and is said to be N -subprojective if for every epimorphism g : B → N and homomorphism f : M → N , there exists a homomorphism h : M → B such that gh = f . For a module M , the subprojectivity domain of M is defined to be the collection of all modules N such that M is N -subprojective. We consider, for every ring R, the subprojective profile of R, namely, the class of all subprojectivity domains for R modules. We show that the subprojective profile of R is a semilattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (pindigent) and those with co-atomic subprojectivy domain are said to be maximally subprojective. While we do not know if sp-poor modules and maximally subprojective modules exist over every ring, their existence is determined for various families. For example, we determine that artinian serial rings have sp-poor modules and attain the existence of maximally subprojective modules over the integers and for arbitrary V-rings. This work is a natural continuation to recent papers that have embraced the systematic study of the injective, projective and subinjective profiles of rings.

An alternative perspective on pure-projectivity of modules

“…p-indigent modules were introduced and some results about them were obtained in [12]. Recall that a module M is called p-indigent (or is called sp-poor in [12]) if subprojectivity domain of M consists of only projective modules.…”

“…In contrast to the notion of relative projectivity, Holston et al introduced in [12] the notion of subprojectivity. Namely, a module M is said to be N-subprojective if for every epimorphism g ∶ B → N and homomorphism f ∶ M → N , then there exists a homomorphism h ∶ M → B such that gh = f .…”

“…So, the smallest possible subprojectivity domain is the class of projective modules. A module with such a subprojectivity domain is defined in [12] to be p-indigent. P-indigent modules studied further in [6] where Durgũn investigated the structure of rings over which every module is p-indigent or projective.…”

“…Inspired by the notion of relative subprojectivity from [12], in this paper we initiate the study of an alternative perspective on the analysis of the pure-projectivity of a module, as we introduce the notions of relative pure-subprojectivity and assign to every module its pure-subprojectivity domain. The aim of this paper is to investigate the viability of obtaining valuable information about a ring R from the perspective of pure-subprojectivity domain.…”

Subprojectivity domains of pure-projective modules

“…The projective analog of indigent modules was considered in [12], namely, pindigent modules. A module M is p-indigent if Pr −1 (M ) consists precisely of the projective modules.…”