Rings Whose Cyclic Modules Are ⊞-Supplemented (Adjunct) Modules

Abstract

This paper introduces and systematically investigates the class of  -adjunct modules, a generalization of adjunct modules defined by the existence of direct summand complements with a certain small intersection property. We establish sufficient conditions for direct summands and factor modules of a -adjunct modules to inherit this property. Our primary focus is the characterization of rings over which every cyclic right module is -adjunct modules. We demonstrate that such rings are necessarily quasi-flawless and provide a precise internal ring-theoretic characterization. Furthermore, we distinguish this new class by constructing examples showing it lies strictly between the rings whose cyclic modules are ascending and those whose cyclic modules are merely adjunct, and prove that this property is not left-right symmetric.