Projectivity and quasi-projectivity with respect to epimorphisms to simple modules

Using the notion of relative max-projectivity, max-projectivity domain of a module is investigated. Such a domain includes the class of all modules whose maximal submodules are direct summands (this class denoted as [Formula: see text]). We call a module max-[Formula: see text]-poor if its max-projectivity domain is exactly the class [Formula: see text]. We establish the existence of max-[Formula: see text]-poor modules over any ring. Furthermore, we study commutative rings whose simple modules are projective or max-[Formula: see text]-poor. Additionally, we determine the right Noetherian rings for which all right modules are projective or [Formula: see text]-poor. Max-[Formula: see text]-poor abelian groups are fully characterized and shown to coincide precisely with [Formula: see text]-poor abelian groups. We also further investigate modules that are max-projective relative to themselves, which are known as simple-quasi-projective modules. Several properties of these modules are provided, and the structure of certain classes of simple-quasi-projective modules is determined over specific commutative rings including the ring of integers and valuation domains.