On τ -lifting Modules and τ -semiperfect Modules

Motivated by [1], we study on τ -liftingmodules (rings) and τ -semiperfect modules (rings) for a preradical τ and give some equivalent conditions. We prove that; i) if M is a projective τ -lifting module with τ(M) ⊆ δ(M), then M has the finite exchange property; ii) if R is a left hereditary ring and τ is a left exact preradical, then every τ -semiperfect module is τ –lifting; iii) R is τ -lifting if and only if every finitely generated free module is τ -lifting if and only if every finitely generated projective module is τ -lifting; iv) if τ(R) ⊆ δ(R), then R is τ -semiperfect if and only if every finitely generated module is τ -semiperfect if and only if every simple R–module is τ -semiperfect.