On the Torsion-Theoretical Aspect of Supplementing Modules

In this paper, we intended to extend some well-known results for the torsion-theoretical generalization of supplementing modules. A module that has a tau-supplement in every extension is called tau-supplementing modules, where tau is an idempotent radical. We study these modules by generalizing several results both on supplementing modules and injective modules. We study generalizations and characterizations of SI-ring, semiartinian rings and perfect rings by tau-supplementing modules.