R-Annihilator - Hollow Modules

Let R be an associative ring with identity and let M be a unitary left R- module. We call a non-zero module M, R-annihilator hollow module if every proper submodule of M is R-annihilatorsmall submodule of M. . The sum A_M of all such submodules of M contains the Jacobson radical J (M) and the singular submodule Z (M). When M is finitly generated and faithful, we study A_M and K_M in this paper. Conditions when A_M is R-annihilator-small and K_M= A_ (M), J (M) A_Mand Z (M) A_M are given.