In [9], the author extends the definition of lifting and supplemented modules to

-lifting and

-supplemented by replacing "small submodule" with "

-small 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.