A new radical in free modules

Document Type : Research Paper

Authors

Department of Mathematics, Hormozgan University, Bandar Abbas, Iran.

Abstract

R

-module

M

is called torsion-free, if

r x = 0

for

r \in R

and

x \in M

implies that

r = 0

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

\frac{R}{P}

-module

\frac{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} ⫋ F

, we have for any

r \in R

and

m \in F

r m \in N

implies

r \in \sqrt{P}

m \in \sqrt{N}

if and only if

\frac{R}{P}

-module

\frac{F}{N}

is quasi torsion free.

Keywords