An $R$-module $M$ is called torsion-free, if $rx=0$ for $r\in R$ and $x\in M$ 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 $\dfrac{R}{P}$-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.
Adhami, P., & Moghaderi, J. (2022). A new radical in free modules. Journal of Algebra and Related Topics, 10(1), 71-94. doi: 10.22124/jart.2022.20214.1295
MLA
P. Adhami; J. Moghaderi. "A new radical in free modules". Journal of Algebra and Related Topics, 10, 1, 2022, 71-94. doi: 10.22124/jart.2022.20214.1295
HARVARD
Adhami, P., Moghaderi, J. (2022). 'A new radical in free modules', Journal of Algebra and Related Topics, 10(1), pp. 71-94. doi: 10.22124/jart.2022.20214.1295
VANCOUVER
Adhami, P., Moghaderi, J. A new radical in free modules. Journal of Algebra and Related Topics, 2022; 10(1): 71-94. doi: 10.22124/jart.2022.20214.1295
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