Characterizations of Lifting Modules in Terms of Cojective Modules and the Class of ℬ(m,x)

Abstract

In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small, pseudo-)cojective modules via the class ℬ(M,X). Let M = M1 ⊕ M2 be an X-amply supplemented module with the finite internal exchange property. Then for every decomposition of M = Mi ⊕ Mj, Mi is ℬ(Mj,X)-cojective for i ≠ j, M1 and M2 are X-lifting if and only if M is X-lifting. We also prove that for an X-amply supplemented module M = M1 ⊕ M2 such that M1 and M2 are indecomposable X-lifting modules, if M2 is ℬ(M1,X)-cojective and M1 is small-ℬ(M2,X)-cojective then M is X-lifting.

Lifting Modules

“…Using this we prove equivalent conditions for a finite direct sum of hollow modules to be lifting in 23. 16 [30]; Hanada,Kuratomi and Oshiro in [143]; Harada [156]; Harada and Mabuchi [161]; Harada and Tozaki [163]; Keskin [199]; Kuratomi [214]; Mohamed and Müller [244,245]; Orhan and Keskin Tütüncü [268].…”

“…[61]; Hanada, Kado and Oshiro [143]; Kuratomi [214]; Kutami and Oshiro [215]; Miyashita [235]; Mohamed, Müller and Singh [246]; Orhan and Keskin Tütüncü [268]. ; As lifting modules and self-projective modules are closed under direct summands, any direct summand of a strongly discrete module is strongly discrete.…”