A Module Whose Second Spectrum Has the Surjective or Injective Natural Map

Abstract

‎Let $R$ be a commutative ring and $M$ be an $R$-module‎. ‎Let $Spec^{s}(M)$ be the set of all second submodules of $M$‎. In this article‎, ‎we topologize $Spec^{s}(M)$ with Zariski and classical Zariski topologies and study the classes of all modules whose second spectrum have the surjective or injective natural map‎. ‎Moreover‎, ‎we investigate the interplay between the algebraic properties of $M$ and the topological properties of $Spec^{s}(M)$‎.