e-lifting modules relative to fully invariant submodules

In this paper, we introduce the notion FI-e-lifting modules which is a proper generalization of lifting (e-lifting) modules.Then we give some characterizations and properties of e-lifting and FI-e-lifting modules. We provide a decomposition of any e-lifting modules. It is shown that every finite direct sum of FI-e-lifting modules is FI-e-lifting