Second Spectrum of Modules and Spectral Spaces

Abstract

Let R be a commutative ring with identity and \(\hbox {Spec}^{s}(M)\) denote the set all second submodules of an R-module M. In this paper, we investigate various properties of \(\hbox {Spec}^{s}(M)\) with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for \(\hbox {Spec}^{s}(M)\) to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop.