Zariski topology on the secondary-like spectrum of a module

Abstract

Abstract Let ℜ \Re be a commutative ring with unity and ℑ \Im be a left ℜ \Re -module. We define the secondary-like spectrum of ℑ \Im to be the set of all secondary submodules K K of ℑ \Im such that the annihilator of the socle of K K is the radical of the annihilator of K K , and we denote it by Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) . In this study, we introduce a topology on Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Spec s ( ℑ ) {{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.