Ceken, Secil2024-06-122024-06-1220200354-5180https://doi.org/10.2298/FIL2002483Chttps://hdl.handle.net/20.500.14551/237551st Mediterranean International Conference of Pure and Applied Mathematics and Related Areas (MICOPAM) -- OCT 26-29, 2018 -- Akdeniz Univ, Antalya, TURKEYLet 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 is an element of Spec(s)(M) : ann(R)(S) = p g. A second submodule Q ofMis called an upper second submodule if there exists a prime ideal p of R such that Spec(p)(s)(M)not equal (sic) and Q = Sigma S is an element of Spec(p)(s)(M) S. The set of all upper second submodules ofMis called upper second spectrumofMand denoted by u:Specs(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:Specs(M) with the patch topology and the finer patch topology. We show that for every left R-moduleM, 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:Specs(M) with the dual Zariski topology is a spectral space.en10.2298/FIL2002483Cinfo:eu-repo/semantics/openAccessSecond SubmoduleUpper Second SubmoduleDual Zariski TopologyPatch TopologySpectral SpacePrime Spectrum2nd SpectrumModuleNotionConference Object342483489Q3WOS:0005953297000222-s2.0-85096963623Q3