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.