On the prime spectrum of an le-module

Abstract

Here, we continue to characterize a recently introduced notion, le-modules [Formula: see text] over a commutative ring [Formula: see text] with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec[Formula: see text] of all prime submodule elements of [Formula: see text]. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec[Formula: see text] is connected if and only if [Formula: see text] contains no idempotents other than [Formula: see text] and [Formula: see text]. Open sets in the Zariski topology for the quotient ring [Formula: see text] induces a base of quasi-compact open sets for the Zariski topology on Spec[Formula: see text]. Every irreducible closed subset of Spec[Formula: see text] has a generic point. Besides, we prove a number of different equivalent characterizations for Spec[Formula: see text] to be spectral.