An $R$-module $M$ is defined to satisfy epi-retractability on ascending (resp . descending) chains of submodules if in every ascending (resp . descending) chain $\\\\{M_{i}\\\\}$ of submodules of $M$ , there exists $k\\\\in{\\\\mathbb{N}}$ such that for each $i\\\\geq k$ , there exists an epimorphism $M_{i+1}\\\\longrightarrow M_{i}$ ($M_{i}\\\\longrightarrow M_{i+1}$) . In this thesis , we study these modules . We obtain some results about these modules in general . Then we investigate some special cases such as infinite direct sums of modules . We characterize semiprime right Goldie rings with epi-retractability on descending chains of right ideals . We get some necessary conditions for a ring $R$ such that all $R$-modules satisfy epi-retractability on ascending (resp . descending) chains of submodules . We also consider a particular type of these modules , namely modules with divisibility on chains of submodules . We say that an $R$-module $M$ satisfies divisibility on ascending (resp . descending) chains of submodules if for every ascending (resp . descending) chain of submodules of $M$ , there exists $k\\\\in{\\\\mathbb{N}}$ such that for each $i\\\\geq k$ , there exists an endomorphism of $M$ that sends $M_{i+1}$ ($M_{i}$) onto $M_{i}$ ($M_{i+1}$) . In particular , We study rings with these properties . We consider right self injective regular rings with divisibility on ascending or descending chains of right ideals . We also study right duo right perfect rings with divisibility on descending chains of right ideals . We investigate some injective nonsingular modules with divisibility on ascending or descending chains of submodules . Finally , we study chains of prime submodules in modules with divisibility on ascending or descending chains of submodules and prime modules with these properties .
