Abstract
Let R be a commutative ring with identity, S be a multiplicatively closed subset of R. A submodule N of an R-module M with ann\(_{R}(N)\cap S=\emptyset \) is called an S-second submodule of M if there exists a fixed \(s\in S\), and whenever \(rN\subseteq K\), where \(r\in R\) and K is a submodule of M, then either \(rsN=0\) or \(sN\subseteq K\). The set of all S-second submodules of M is called S-second spectrum of M and denoted by S-\(Spec^{s}(M)\). In this paper, we construct and study two topologies on S-\(Spec^{s}(M)\). We investigate some connections between algebraic properties of M and topological properties of S-\(Spec^{s}(M)\) such as seperation axioms, compactness, connectedness and irreducibility.
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References
Abuhlail, J.Y.: Zariski topologies for coprime and second submodules. Algebra Coll. 22(1), 47–72 (2015)
Anderson, D.D., Dumitrescu, T.: S-noetherian rings. Commun. Algebra 30(9), 4407–4416 (2002)
Ansari-Toroghy, H., Farshadifar, F.: The dual notion of multiplication modules. Taiwan. J. Math. 11(4), 1189–1201 (2007)
Ansari-Toroghy, H., Farshadifar, F.: The Zariski topology on the second spectrum of a module. Algebra Coll. 21(4), 671–688 (2014)
Çeken, S., Alkan, M.: Dual of Zariski topology for modules. Book Ser. AIP Conf. Proc. 1389(1), 357–360 (2011)
Çeken, S., Alkan, M., Smith, P.F.: The dual notion of the prime radical of a module. J. Algebra 392, 265–275 (2013)
Çeken, S., Alkan, M., Smith, P.F.: Second modules over noncommutative rings. Commun. Algebra 41(1), 83–98 (2013)
Dauns, J.: Prime submodules. J. Reine Angew. Math. 298, 156–181 (1978)
Farshadifar, F.: S-secondary submodules of a module. Commun. Algebra 49(4), 1394–1404 (2021)
Farshadifar, F.: S-second submodules of a module. Algebra and Discrete Mathematics 32(2), 197–210 (2021)
Fuchs, L., Heinzer, W.J., Olberding, B.: Commutative ideal theory without finiteness conditions: completely irreducible ideals. Trans. Am. Math. Soc. 358, 3113–3131 (2006)
Hamed, A.: S-noetherian spectrum condition. Commun. Algebra 46(8), 3314–3321 (2018)
Hamed, A., Hizem, S.: Modules satisfying the S-Noetherian property and S-ACCR. Commun. Algebra 44, 1941–1951 (2016)
Hamed, A., Malek, A.: S-prime ideals of a commutative ring. Beitrage zur Algebra und Geometrie/Contributions to Algebra and Geometry 61, 533–542 (2020)
Sharp, R.Y.: Steps in Commutative Algebra. Cambridge University Press, Cambridge (2000)
Sevim, E.S., Arabaci, T., Tekir, Ü., Koç, S.: On S-prime submodules. Turk. J. Math. 43(2), 1036–1046 (2019)
Sevim, E.S., Tekir, Ü., Koç, S.: S-artinian rings and finitely S-cogenerated rings. J. Algebra. Appl. 19(03), 2050051 (2020)
Yassemi, S.: The dual notion of prime submodules. Arch. Math (Brno) 37, 273–278 (2001)
Yıldız, E., Ersoy, B.A., Tekir, Ü., Koç, S.: On S-Zariski topology. Commun. Algebra 49(3), 1212–1224 (2021)
Yıldız, E., Tekir, Ü., Koç, S.: On S-comultiplication modules. Turk. J. Math. 46, 2034–2046 (2022)
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Çeken, S. On S-second spectrum of a module. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 171 (2022). https://doi.org/10.1007/s13398-022-01316-3
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DOI: https://doi.org/10.1007/s13398-022-01316-3