Given a prime number p, we study the module theory of F[G], where F is a field of characteristic p and G is a cyclic p-group. We describe a construction of the set of all injective homomorphisms between two finitely generated F[G]-modules in terms of their numerical invariants. We also give a conceptual characterization of injective F[G]-homomorphisms. Finally, we characterize all submodules of a given finitely generated F[G]-module. These results were applied to describe all solutions of a specific type of Galois embedding problems in [8].