Through this paper R is an associative ring with identity and all modules are unitary right R-modules. A submodule N of a module M is called small in M (denoted by Nl−/M) if for every proper submodule L of M, N + L = M. M is called a small module if M is small in some modules [1]. In [1] Leonard has proved that a module M is a small module if and only if it is small in its injective hull. For a module M let Z(M) = Rej(M, S) = {Kerf | f : M → S, S ∈ S} = {U ⊆ M | M/U ∈ S} where S denotes the class of all small modules. M is called cosingular if Z(M) = 0. It is obvious that every small module is cosingular but in general the converse is not true. A module M has C∗ if every submodule N of M contains a direct summand K of M such that N/K is cosingular. Finally we recall that a module M is lifting if every submodule N of M contains a direct summand K of M such that N/Kl−/M/K.
