On the prime spectrum of an le-module

TL;DR

This paper extends the Zariski topology to the set of prime submodule elements of le-modules over a ring, characterizing its topological properties and conditions for spectrality.

Contribution

It introduces and characterizes the Zariski topology on Spec(M) for le-modules, extending classical module results to this new setting.

Findings

Spec(M) is connected iff R/Ann(M) has no nontrivial idempotents.

Open sets in Spec(M) form a basis of quasi-compact open sets.

Every irreducible closed subset has a generic point.

Abstract

Here we continue to characterize a recently introduced notion, le-modules $_{R}M$ over a commutative ring $R$ with unity \cite{Bhuniya}. This article introduces and characterizes Zariski topology on the set $Spec(M)$ of all prime submodule elements of $M$. Thus we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec(M) is connected if and only if $R/Ann(M)$ contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring $R/Ann(M)$ induces a base of quasi-compact open sets for the Zariski-topology on Spec(M). Every irreducible closed subset of Spec(M) has a generic point. Besides, we prove a number of different equivalent characterizations for Spec(M) to be spectral.