In this thesis all rings are associative with identity and all modules are unitary left modules. Let R be any ring. A submodule K of an R -module M is called closed (in M ) provided K has no proper essential axtension in M. Clearly, every direct summand of M is closed in M. Moreover, if L is any submodule of M then there exists a submodule K of M maximal with respect to the property that L is essential submodule of K and this case K is closed submodule of M . A module M is called an extending module if every closed submodule is direct summand. An R- module X is called M-c-injective provided, for every closed submodule K of M, every homomorphism ?: K ? X can be lifted to a homomorphism ?: M ? X. Moreover, X called c-injective provided X is M-c- injective for every R- module M . Not that if M is an extending module then every R- module is M-c- injective. If R is a Dedekind domain and an R- module M is a direct product of simple R- modules then M is M-c -injective. We prove that if R is a Dedekind domain that an R -module X is c -injective if and only if there exists an R- module Y such that Y is a direct product of simple R -modules and injective R -modules with the property that X is isomorphic to a direct summand of Y. We show that such a direct summand is isomorphic to a direct product of homogeneous semisimple R- modules and injective R -modules. Let P is collection of left primitive ideals of R . A submodule L of an R -module M is called P -pure in M provided L ? IM=IL for every ideal I in P . An R -module X is called P -pure-injective provided, for every R -module and every P -pure submodule L of M , every homomorphism ?: L ? X can be lifted to a homomorphism ?: M ? X . We first characterize P - pure-injective modules over rings R such that R ? P is an Artinian ring for every left primitive ideal P of R . Commutative rings clearly have this property. More generally rings satisfying a polynomial identity, FBN rings and semiperfect rings satisfy this property. Then show that for a Dedekind domain the class of c -injective modules is precisely the class of P -pure-injective modules. We show that the characterization does not extend to commutative Noetherian domains which are not Dedekind. In fact we prove that if R is a commutative Noetherian domain and P a maximal ideal of R then the simple R -module R ? P is c -injective if and only if the ideal P is invertible. Finally we show that a commutative Noetherian domain R is Dedekind if and only if every simple R -module is c -injective.