Modules in σ[M] with Chain Conditions on δ_M^-small Submodules

Problem statement: Let M be a right module over a ring R. In this article modules in σ[M] with chain conditions on δM- small submodules are studied. Approach: With the help of known results about M- singular, Artinian and Noetherian modules the techniques of the proofs of our main results use the properties of δ_M- small, δ_M- supplement and δ_M- semimaximal submodules. Results: Modules in σ[M] with chain conditions on δ_M- small are investigated, δ_M- semimaximal submodule is defined. Some Properties of δ_M- semimaximal submodules are proved. As application a new characterization of Artinian module in σ[M] is obtained in terms of δ_M- small submodules and δ_M- semimaximal submodules, as well as δM- small submodules and δM- supplement submodules. Conclusion/Recommendations: Our results certainly generalized several results obtained earlier.