On lifting modules

Let R be a ring with identity and let M = M(1)circle plus...circle plus M-n , be a finite direct sum of relatively projective R-modules M-i. Then it is proved that M is lifting if and only if M is amply supplemented and M-i is lifting for all 1 less than or equal to i less than or equal to n. Let M - M(1)circle plus...circle plus M-n be a finite direct sum of R-modules Mi. We prove that M is (quasi-) discrete if and only if M1, -, M, are relativel y projective (quasi-) discrete modules. We also prove that, for an amply su pplemented R-module M = M(1)circle plus M-2 such that M-1 and M-2 have the finite exchange property, M is lifting if and only if n M-1 and M2 are lift ing 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 prov e that, for a ring R such that every direct sum of a lifting R-module and a simple R-module is lifting, every simple R-module is small M-projective fo r any lifting R-module M.