Abstract Let R be a ring and M a right R-module. We call a module M is FI-δ-lifting if every fully invariant submodule A of M contains a direct summand B of M such that A/B «δ M/B. In this paper several properties of these modules are studied. We show that a ring R is FI-δ-lifting as an R-module if and only if R/I has a projective δ-cover for every two sided ideal I of R.
