Lifting modules, extending modules and their applications to generalized uniserial rings

Artinian serial principal ideal rings and artinian serial rings are traditionally called uniserial rings and generalized uniserial rings, respectively.These rings are important classical artinian rings as well as quasi-Frobenius rings.The reader is referred to Faith's Book [4] for these rings.As is well known, a ring R is a quasi-Frobenius ring iff every injective R-module is projective, and if every projective R module is injective; while R is a uniserial ring iff every quasi-injective R-module is quasi-projective, and iffThe purpose of this paper is to give similar characterizations of a gen- eralized uniserial ring R in terms of extending and lifting modules.More specifically, consider the following implications: injective quasi-injective extending As just noted above, R is quasi-Frobenius⇔ a ) ⇔ a* ); while R is uniserial0&) ⇔ b*) .The conditions d) and d* ) are recently studied by Harada ([6]~[8]) and Oshiro ([15]).In this paper, we study c), c* ), e) and e* ) and show the following result: R is generalized uniserial⇔ e ) ⇔ e* ) ⇔ c ) ⇔ R is a right perfect ring with c* ).NOTATION.Throughout this paper, we assume that R is an associative ring with identity and all R -modules are unitary right R modules.Let M be an R -module.