Zariski topology on the secondary-like spectrum of a module

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 KK of ℑ\Im such that the annihilator of the socle of KK is the radical of the annihilator of KK, and we denote it by SpecL(ℑ){{\rm{Spec}}}^{L}\left(\Im ). In this study, we introduce a topology on SpecL(ℑ){{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Specs(ℑ){{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.

Keywords

• secondary-like spectrum
• zariski topology
• second spectrum
• 13c13
• 13c99
• 54b99