A new radical in free modules

An R-module M is called torsion-free, if rx=۰ for r\in R and x\in M implies that r=۰ or x=۰. 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 \dfrac{R}{P}-module \dfrac{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 \sqrt N \subsetneqq F, we have for any r \in R and m \in F, rm \in N implies r \in \sqrt P or m \in \sqrt N if and only if \dfrac{R}{P}-module \dfrac{F}{N} is quasi torsion free.

Keywords:

prime submodule , primary submodule , Free module