Keskin and Harmanci defined the family B (M, X)= ${A {\leq} M|{\exists} Y {\leq} X,{\exists} f
{\in} Hom_R (M, X/Y),\; Ker\; f/A {\ll} M/A} $. And Orhan and Keskin generalized projective …

Let R be a ring with identity and let $ M= M_1 {\bigoplus} M_2 $ be an amply supplemented
R-module. Then it is proved that $ M_i $ has ($ D_1 $) and is $ M_j-^* ojective $ for $ i {\neq} …

A module M is called extending if, for any submodule X of M, there exists a direct summand
of M which contains X as an essential submodule, that is, for any submodule X of M, there …

A submodule N of a right R-module M is called coneat if for every simple right R-module S,
any homomorphism $ N {\rightarrow} S $ can be extended to a homomorphism $ M …

Let $ R $ be any ring and let $ M $ be any right $ R $-module. $ M $ is called hollow-lifting if
every submodule $ N $ of $ M $ such that $ M/N $ is hollow has a coessential submodule …

A submodule N of a module M is called ${\mathcal {S}} $-closed (in M) if M/N is nonsingular.
It is well-known that the class Closed of short exact sequences determined by closed …

In [9, Theorem 3.1] KR Fuller characterized indecomposable injective projective modules
over artinian rings using i-pairs. In [3] the author generalized this theorem to …

A module M over a ring R is called hollow-lifting if every submodule N of M with M/N hollow
contains a direct summand K of M such that N/K is a small submodule of M/K. It is known that …

J. Korean Math. Soc. 46 (2009), No. 1, pp. 51–58 Page 1 J. Korean Math. Soc. 46 (2009), No. 1,
pp. 51–58 THE CLASS OF MODULES WITH PROJECTIVE COVER YONGHUA GUO …

In this paper, we study a module which is lifting and supplemented relative to a module
class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules …