On $tau$-lifting modules and $tau$-semiperfect modules

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

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