On lifting modules

Keskin D.

COMMUNICATIONS IN ALGEBRA, vol.28, no.7, pp.3427-3440, 2000 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 28 Issue: 7
  • Publication Date: 2000
  • Doi Number: 10.1080/00927870008827034
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.3427-3440
  • Hacettepe University Affiliated: Yes


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 relatively projective (quasi-) discrete modules. We also prove that, for an amply supplemented 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 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 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 for any lifting R-module M.