X-Lifting Modules Over Right Perfect Rings

S -FP-Projective Modules and Dimensions

“…From reference [1], we recall that an R-module M is said to be u-S-torsion if sM � 0 for some s ∈ S. An R-module M is said to be S-fnite if M/F is u-S-torsion for some f.g. submodule F of M. Also, following Zhang [1,2], a sequence 0 ⟶ A ⟶ β B ⟶ c C ⟶ 0 is said to be u-S-exact (at B) provided that there is an element t ∈ S such that tKer(c)⊆Im(β) and tIm(β)⊆Ker(c). A long R-sequence .…”

“…Recently, Zhang in reference [1] defned the u-S-von Neumann regular ring as follows: A ring R is called a u-S-von Neumann regular ring if there exists an element s ∈ S such that for any a ∈ R, there exists r ∈ R with sa � ra 2 . Tus, by reference [1], Teorem 3.13, R is a u-S-von Neumann regular ring if and only if every R-module is u-S-fat, and in reference [6] Teorem 3.5, it is proved that a ring R is u-S-von Neumann regular if and only if every R-module is u-S-absolutely pure.…”

On the Decomposition of Extending Lifting Modules

“…Recently Chang showed that if every co-closed submodule of any projective module P contains Rad(P ), then every X-lifting module over a right perfect ring has an indecomposable decomposition (cf. [1,3,7,12]). …”