Rings whose modules have maximal or minimal subprojectivity domain

Abstract

Given modules M and N, 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. As an alternative perspective on the projectivity of a module, a module M is said to be p-indigent if its subprojectivity domain is smallest possible, namely, consisting of exactly the projective modules. Properties of subprojectivity domains and of p-indigent modules are studied. For various classes of modules (such as simple and singular), necessary and sufficient conditions for the existence of p-indigent modules of those types are studied. We characterize the rings over which every (simple) module is projective or p-indigent. In addition, we use our results to provide a characterization of a special class of QF-rings in which the subinjectivity and subprojectivity domains of all modules coincide. As the projective analog of indigent modules, p-indigent modules were introduced by Holston, López-Permouth, Mastromatteo and Simental-Rodriguez. The paper is inspired by similar ideas and problems in papers by Aydoğdu and López-Permouth and by Alizade, Büyükaşık and Er, where an injective version of p-indigent modules is introduced and studied.