Modules satisfying certain chain conditions and their endomorphisms

In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M , every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M , every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M , EndRM is a division ring.