On lifting modules

Let R be a ring with identity and let be a finite direct sum of relatively protective R-modules Mi Then it is proved that M is lifting if and only if M is amply supplemented and Mi is lifting for all 1 ≤ i ≤ n.Let be a finite direct sum of R-modules Mi . We prove that M is (quasi-) discrete if and only if are relatively projective (quasi-) discrete modules. We also prove that, for an amply supplemented R-module M=M 1⊕M 2 such that M 1 and M 2 have the finite exchange propertyM is lifting if and only if M 1 and M 2 are lifting and relatively small projective R-modules and every co-closed submodule N of M with M= N+M 1 = N+M 2 is a direct summand of M.Finally, we prove that, for a ring Rsuch that every direct sum of a lifting R-module and a simple R-module is lifting, every simple R-module is small M-projective for any lifting R-module M.