# On the Spectrum of Weakly Prime Submodule

## Jituparna Goswami, Helen K. Saikia

## Keywords:

weakly prime submodule, zariski topology, weakly prime radical, flat module## Abstract

A proper submodule $P$ of an $R$-module $M$ is called a weakly prime submodule, if for each submodule $K$ of $M$ and elements $a$, $b$ of $R$, $abK \subseteq P$ implies that $aK \subseteq P$ or $bK \subseteq P$. Let $WSpec(M)$ be the set of all weakly prime submodules of $M$. In this paper, a topology on $WSpec(M)$ is introduced. We investigate some basic properties of the open and closed sets in that topology and establish their relationships with weakly prime radical and Flat Module. We also investigate some topological properties in $WSpec(M)$ such as connectedness, separation axioms etc. Finally we try to characterize the spectrum of weakly prime submodule with the help of quasi multiplication module. we prove that if $M$ is a finitely generated quasi multiplication $R$-module then $WSpec(M)$ is compact.## Downloads

## Published

2021-03-01

## How to Cite

*Thai Journal of Mathematics*,

*19*(1), 51–58. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1134

## Issue

## Section

Articles