On t-lifting Modules and t-semiperfect Modules

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

On t-lifting Modules and t-semiperfect Modules

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