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

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