On the Upper Dual Zariski Topology

Abstract

Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Spec^{s}(M). For each prime ideal p of R we define Spec_{p}^{s}(M):={S∈Spec^{s}(M):ann_{R}(S)=p}. A second submodule Q of M is called an upper second submodule if there exists a prime ideal p of R such that Spec_{p}^{s}(M)≠∅ and Q=∑_{S∈Spec_{p}^{s}(M)}S. The set of all upper second submodules of M is called upper second spectrum of M and denoted by u.Spec^{s}(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u.Spec^{s}(M) with the dual Zarsiki topology. Also, we topologize u.Spec^{s}(M) with the patch topology and the finer patch topology. We show that for every left R-module M, u.Spec^{s}(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Spec^{s}(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster's characterization of a spectral space, we show that if M is an Artinian left R-module, then u.Spec^{s}(M) with the dual Zariski topology is a spectral space.