a new radical in free modules

An r-module m is called torsion-free, if rx=0 for r ∈ r and x ∈ m implies that r=0 or x=0. in this paper, we introduce the notions semi torsion-free modules and quasi torsion-free modules. we show that a submodule n of an r-module m is a p-primary submodule if and only if r/p-module m/n is semi torsion-free. also we define a new radical in free modules and find some characterizations of it. we prove that for p-submodule n of a free r-module f which √ n ⊂≠ f, we have for any r ∈ r and m ∈ f, rm ∈ n implies r ∈ √ p or m ∈ √ n if and only if r/p-module f/n is quasi torsion free.