On subprojective portfolios of modules

The study of subprojective profiles on module categories is closely related to the study of (co)torsion theories. We study in this article the collection of subprojective portfolios, which is referred to as the subprojective profile (shortly, sp-profile) of a ring R. We study some properties of the sp-profile and consider ways in which properties of this profile may determine the structure of rings and vice versa. We investigate the case when the sp-profile of a ring form a lattice. We further study in detail the situation when sp-profile of a ring R is a chain.