Modules With Chain Conditions On δ -Small Submodules

Let R be an associative ring with identity and M be unital non zero R-module. A submodule N of a module M is called a δ-small submodule of M (briefly N δ M )if N+X=M for any proper submodule X of M with M/X singular, we have X=M . In this work,we study the modules which satisfies the ascending chain condition (a. c. c.) and descending chain condition (d. c. c.) on this kind of submodules .Then we generalize this conditions into the rings , in the last section we get same results on δ- supplement submodules and we discuss some of these results on this types of submodules.