Let R be a commutative ring with identity and M be a unital R-module. In this thesis there are given a partial characterization of invertible, dense and projective submodules and the equivalent conditions. Also it is provided some conditions for R and M such that the given ring R is a Dedekind domain if and only if every non zero submodule of R-module M is locally free. The relations between a finitely generated torsionfree Dedekind module M over a domain R, prime submodules of the O(M)-module M and the ring O(M) are given. It is proved that M is a finitely generated torsionfree Dedekind module over a domain R if and only if M is a noetherian O(M)-module and every semi-maximal submodule of the R-module M is invertible.