On RD -Projectively Poor Modules and Their Ring-Theoretic Implications

In this paper, the notion of [Formula: see text]-projectively poor ([Formula: see text]-poor) modules is introduced and systematically investigated. These are defined as modules whose [Formula: see text]-projectivity domain is minimal. It is demonstrated that the class of rings over which every module is [Formula: see text]-poor coincides precisely with the class of Köthe rings, thereby providing a new characterization of these rings in terms of minimal [Formula: see text]-projectivity behavior. The structural relationships between [Formula: see text]-poor, [Formula: see text]-poor and pure-projectively poor ([Formula: see text]-poor) modules are explored. In particular, it is proved that the classes of [Formula: see text]-poor and [Formula: see text]-poor modules coincide over several important classes of rings, including [Formula: see text]-rings and Prüfer rings. Moreover, it is proved that [Formula: see text]-poor modules and [Formula: see text]-poor modules coincide over von Neumann regular rings. Additionally, the concept of [Formula: see text]-split modules is defined and their basic properties are established. Finally, a sufficient condition for the existence of [Formula: see text]-poor modules over Prüfer rings is provided.