Abstract

Following (Kosa̧n in Algebra Colloq. 14:53–60, 2007), a module M is called δ-supplemented if, for every submodule of N of M, there exists L≤N such that M=N+L and N∩L≪ δ L. A module M is called δ-lifting if, for any N≤M, there exists a decomposition M=A⊕B such that A≤N and N∩B≪ δ M. In this paper, we study e-supplemented modules and e-lifting modules which are generalized δ-supplemented modules and δ-lifting modules. Some properties of these modules are considered. Moreover, we also have new characterizations of Artinian (resp., Noetherian) Rad e (M) module studied with chain conditions on e-supplemented modules.