AN ALTERNATIVE PERSPECTIVE ON PROJECTIVITY OF MODULES

HOLSTON, CHRIS ; LÓPEZ-PERMOUTH, SERGIO R. ; MASTROMATTEO, JOSEPH ; SIMENTAL-RODRÍGUEZ, JOSÉ E.

Glasgow mathematical journal, Tome 57 (2015) no. 1, pp. 83-99

Résumé

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 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 semi-lattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is as smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (p-indigent), 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.

Détail

DOI : 10.1017/S0017089514000135

Mots-clés : 16D40, 16D50, 16D80