EXTREMITY CONCEPTS OF LIFTING MODULES

Recall that an R-module M is lifting if every submodule of M lies above a direct summand of M. In this paper, we introduce and study the classes of modules which are extremity of lifting modules. We call an R-module M is strongly lifting if every submodule of M lies above a stable direct summand of M. Also, we call R-module M is S-lifting if every stable submodule of M lies above a direct summand of M. In fact, the following proper hierarchy is concluded: Strongly lifting modules → Lifting modules → S-lifting modules Some counter examples are given to separate these concepts. Also, many characterizations and properties of strongly lifting (respectively, S-lifting) modules are obtain. It is shown that a module M is strongly lifting if and only if M is lifting and M is SS-modules. Moreover, we investigate whether the class of strongly lifting (respectively, S-lifting) modules are closed under particular class of submodules, direct summands and direct sums. It shown that a finite direct sum of S-lifting modules is S-lifting.