Some Results on δ-Semiperfect Rings and δ-Supplemented Modules

Abstract

In [9], the author extends the definition of lifting and supplemented modules to -lifting and -supplemented by replacing small submodule with submodule introduced by Zhou in [13]. The aim of this paper is to show new properties of -lifting and -supplemented modules. Especially, we show that any finite direct sum of -hollow modules is -supplemented. On the other hand, the notion of amply -supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply -supplemented and satisfies Descending Chain Condition (DCC) on -supplemented modules and on -small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a -semiperfect ring which satisfies DCC on -small right ideals of R.