The concepts of subinjectivity, the (sub)injectivity domain of an \(R\)-module \(M\), indigent modules and the right middle class, as well as their projective analogs, viz. subprojectivity, the (sub)projectivity domain, \(p\)-indigent modules and the right \(p\)-middle class have been introduced recently as alternative perspectives on injectivity and projectivity. The study of the injective case is relatively advanced; the projective case is taken further in this paper. A module \(M\) is said to be \(N\)-subprojective if \(M\) has the projective property with respect to any short exact sequence \(B\to N\to 0\). The subprojectivity domain of \(M\), denoted \(\mathbf{Pr}^{-1}(M)\), is the collection of all modules \(N\) such that \(M\) is \(N\)-subprojective. A module \(M\) is called \(p\)-indigent if \(\mathbf{Pr}^{-1}(M)\) consists of exactly the projective modules. If \(\mathbf{FP}\) is the direct sum of all representatives of modules of the class of finitely presented modules, then \(\mathbf{Pr}^{-1}(\mathbf{FP})\) is shown to consist of exactly the flat modules. This means that \(R\) is a right perfect ring if and only if \(\mathbf{FP}\) is \(p\)-indigent. Moreover, \(R\) is a QF-ring if and only if there is a \(p\)-indigent module which is contained in a projective module.
A ring \(R\) is said to have no subprojective \(\mathbf A\)-middle class if each element of the class \(\mathbf A\) is either \(p\)-indigent or projective. Characterizations are found for a nonsemisimple ring \(R\) to have no subprojective middle class. Certain Artinian serial rings are shown to have no subprojective middle class and necessary and sufficient conditions are found for a nonsemisimple QF-ring to have no subprojective middle class.
Several equivalent conditions are found for a nonsemisimple ring \(R\) to have no subprojective middle class. This leads to the result that a ring \(R\) which is not von Neumann regular has no subprojective middle class if and only if \(R\) is fully saturated.
A ring \(R\) is said to satisfy property \(P\) if its subinjectivity domains and subprojectivity domains are equal. It is shown that a ring satisfies \(P\) if and only if it is isomorphic to a product of full matrix rings over Artinian chain rings.