Hollow-R-Annihilator-Lifting Modules

Asubmodule N of amodule W is R-annihilator-small submodule iin W, (N ≪ a W), if whenever N+T = W, where T a submodule of W, implies that ann R (T) = 0 where ann R (T) = {r ∊ R; r.T = 0}. AnR-module W is R-annihilator-lifting module if for every submodule N of W there exists submodules L and K ofW such that W = L ⊕ K with L ≤ N and N ∩ K ≪ a K. Amodule W is called hollow-R-annihilator-lifting module, if for every submodule F of W with W F is hollow, there exiists a direct summand K ofW and F K ≪ a W K ( K ≤ ace F in W ) . Inthis paper, we prove various for this kind of module.