On J–Lifting Modules

Abstract

Let R be a ring with identity and M is a unitary left R–module. M is called J–lifting module if for every submodule N of M, there exists a submodule K of N such that M = K ⊕ K ′ , K ′ ⊆ M and N ∩ K ′ ≪ J K ′ . The am of this paper is to introduce properties of J–lifting modules. Especially, we give characterizations of J–lifting modules.We introduce J–coessential submodule as a generalization of coessential submodule . Finally, we give some conditions under which the quotient and direct sum of J–lifting modules is J–lifting.

F-J-semi Regular Modules

“…Let be a cyclic submodule of . since and are hollow , then is hollow and hollow by [9] but is hollow semi regular , then there exists a submodule projective of such that ́ , ́ and ́ , then is semi regular .…”

On Hollow – J–Lifting Modules

“…Orhan , Keskin and Tribak introduced the concept of hollow-lifting modules; An R-module is hollowlifting if for every submodule N of M with is hollow , there exists a direct summand K of M, such that K is a coessential submodule of N in M [1]. Following Kabban and Khalid [2] , an R-module M is J-lifting module if for every submodule N of M , there exists a submodule K of N, such that M = K Ḱ, Ḱ M and N Ḱ Ḱ . Throughout this paper, R will denote arbitrary rings with identity and all R-modules are unitary left R-modules .…”

“…N is called J-small submodule of M (denoted by N M), if whenever M = N + K, K M , such that J( ) = , implies M = K [3] . Let K and N be submodules of M , such that K N M , then K is called J-coessential submodule of N in M (denoted by K N in M ) if [2] . Recall that a submodule N of an R-module M ISSN: 0067-2904…”