Subprojectivity domains of pure-projective modules

In a recent paper, Holston et al. have defined a module M 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. Clearly, every module is subprojective relative to any projective module. 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 pure-projective module M, the subprojective domain of M. We show that the flat modules are the only ones sharing the distinction of being in every single subprojectivity domain of pure-projective modules. Pure-projective modules whose subprojectivity domain is as small as possible will be called pure-projective indigent (pp-indigent). Properties of subprojectivity domains of pure-projective modules and of pp-indigent modules are studied. For various classes of modules (such as simple, cyclic, finitely generated and singular), necessary and sufficient conditions for the existence of pp-indigent modules of those types are studied. We characterize the structure of a Noetherian ring over which every (simple, cyclic, finitely generated) pure-projective module is projective or pp-indigent. Furthermore, finitely generated pp-indigent modules on commutative Noetherian hereditary rings are characterized.