Let R be a commutative ring and M an R-module. In this thesis, we invastigate the set
T(M)= \{m∈M | ∃0≠r∈R , rm=0 }
of torsion elements of M. Also, we invastigate when T(M) is a submodule of M. We show that if M≠T(M), then T(M) is a union of prime submodules of M.
 

In this article, we study the second radical of module over an arbitrary ring R as the dual notion of the prime radical of a module. We give some propoerties of the second radical of some module. We define the notion of m _* systems. We investigate when the second radical of a module M is equal to the socle of M. In particular, we give a charachterization of the socle of a noetherian module over a ringR such that the ring R/Pis right artinian for every right primitive ideal P by using the concept of second radical. We also give a charachterization of right quasi-due ringsby using the second radical of an injective module.