On the upper dual zariski topology
Filomat, cilt.34, sa.2, ss.483-489, 2020 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 34 Sayı: 2
- Basım Tarihi: 2020
- Doi Numarası: 10.2298/fil2002483c
- Dergi Adı: Filomat
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
- Sayfa Sayıları: ss.483-489
- Anahtar Kelimeler: Dual Zariski topology, Patch topology, Second submodule, Spectral space, Upper second submodule
- Trakya Üniversitesi Adresli: Evet
Özet
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 Specs (M). For each prime ideal p of R we define Specsp(M):= {S ∈ Specs (M): annR(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 Specsp(M) ≠ ∅ and Q = S. The set of all upper second S∈Specsp(M) submodules of M is called upper second spectrum of M and denoted by u.Specs (M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u.Specs (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-module M, u.Specs (M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Specs (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.