R-ANNIHILATOR-SUPPLEMENTED SUBMODULES.

Let R be associative ring with identity and let M be unitary left R- module. Let U and V be submodules of an R-module M. We say that V is R-annihilator-supplement of U in M if M=U+V and whenever Y?V and M=U+Y, then ann Y=0.Let M be an R-module. We say that M is R-annihilator -supplemented module if every proper submodule of M has R-a-supplement.Now let U and V be submodules of an R-module M. We say that V is R-annihilator-weak supplement of U in M if M=U+V and U?V?^aM.Let M be an R-module. We say that M is R-annihilator weakly supplemented module if every submodule of M has R-a-weak supplement in M. The sum ? A?_M of all such submodules of M.When M is finitly generated and faithful , we needed to? A?_M in this paper that are relevant to work, when? A?_M is R-annihilator- small submodules. The main purpose of this work is to develop the properties of R-a-supplemented modules and R-a-weak supplemented modules.