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Projectivity and Quasi-Projectivity With Respect To Epimorphisms To Simple Modules
(World Scientific Publ Co Pte Ltd, 2025) Alagoz, Yusuf; Alagöz, Yusuf; Alizade, Rafail; Büyükaşık, Engin; Buyukasik, Engin; Alizade, Rafail; 04.02. Department of Mathematics; 01. Izmir Institute of Technology; 04. Faculty of Science
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 MDMod -R). We call a module max-p-poor if its max-projectivity domain is exactly the class MDMod -R. We establish the existence of max-p-poor modules over any ring. Furthermore, we study commutative rings whose simple modules are projective or max-p-poor. Additionally, we determine the right Noetherian rings for which all right modules are projective or p-poor. Max-p-poor abelian groups are fully characterized and shown to coincide precisely with p-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.

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