R-annihilator-small submodules

Abstract EN

Let R be an associative ring with identity and let M be a unitary left R- module .We call a submodule K of M , R-annihilator-small if K+T=M ,where T is a submodule of M , implies that ann(T)=0, where ann(T) indicates annihilator of T in R .The sum AM 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 AM and KM in this paper .Conditions when AM is R-annihilator-small and KM=AM, J(M) £ AM and Z(M) £ AM are given.