Let R be a commutative ring with identity and let M be a unitary R-module.In this paper, the structure of completely irreducible submodules will be studied and it is proved that a submodule K has a comlpetely irreducible divisor if and only if Soc(M/K) is nontrivial which implies that a maximal ideal m is an strongly Bourbaki associated prime ideal of K if and only if K has an m-primal completely irrducible divisor. Submodules of M that are representable as an irredundant intersection of an overfamily of completely irreducible submodules are characterized. Then it will be shown that, if R is a Noetherian ring, then M is Artinian if and only if its zero submodule has a primary decomposition whose components are completely irreducible submodules. Finally, it is proved that M is distributive if and only if the set of its completely irreducible submodules is: 
{ m(Rx)_(m) | x ∈ M , m ∈ Max(R) ∩ Supp(Rx) }.