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

Abstract

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