On the Spectrum of Weakly Prime Submodule

Jituparna Goswami, Helen K. Saikia

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.