On projectivity of finitely generated modules

In a recent paper of Holston, Lopez-Permouth, Mastromatteo and Simental-Rodriguez, a ring R is defined to have no subprojective middle class if the subprojectivity domain of any R-module is the smallest or largest possible. In this work, we continue to use this idea of restricting the class of subprojectivity domains to classify rings. A finitely generated (resp., cyclic) module is called fingp-indigent (resp., singp-indigent), if its subprojectivity domain consists of only finitely projective (resp., singly projective) modules. We give a characterization of rings over which finitely generated (resp., cyclic) modules are either projective or fingp-indigent (resp., singp-indigent).