Sonic variations of projectivity

We prove that a ring R has a module M whose domain of projectivity consists of only some injective modules if and only if R is a right noetherian right V-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor R-modules is closed under direct summands if and only if R is a right Bass ring. A ring R is said to have no right max-p-middle class if every right R-module is either projective or max-poor. It is shown that if a commutative noetherian ring R has no right max-p-middle class, then R is the ring direct sum of a semisimple ring R-1 and a ring R-2 which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field Q(2) of R-2 has a proper R-2-submodule which is not complete in its R-2-topology. Then we show that a commutative noetherian hereditary ring R has no right max-p-middle class if and only if R is a semisimple ring.