Korean J. Math. Vol. 28 No. 3 (2020) pp.439-447
DOI: https://doi.org/10.11568/kjm.2020.28.3.439

Second classical Zariski topology on second spectrum of lattice modules

Main Article Content

Pradip Girase
Vandeo C Borkar
Narayan Phadatare

Abstract

Let $M$ be a lattice module over a $C$-lattice $L$. Let $Spec^{s}(M)$ be the collection of all second elements of $M$. In this paper, we consider a topology on $Spec^{s}(M)$, called the second classical Zariski topology as a generalization of concepts in modules and investigate the interplay between the algebraic properties of a lattice module $M$ and the topological properties of $Spec^{s}(M)$. We investigate this topological space from the point of view of spectral spaces. We show that $Spec^{s}(M)$ is always $T_{0}-$space and each finite irreducible closed subset of $Spec^{s}(M)$ has a generic point.



Article Details

References

[1] F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commu- tative ideal theory, Period. Math. Hungar. 30 (1995) (1), 1–26. Google Scholar

[2] E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Dam- ascus Univ. Basic Sci. 19 (2003) (2), 9–21. Google Scholar

[3] H. Ansari-Toroghy and F. Farshadifar, The Zariski topology on the second spec- trum of a module, Algebra Colloq. 21 (2014) (4), 671–688. Google Scholar

[4] H. Ansari-Toroghy, S. Keyvani and F. Farshadifar, The Zariski topology on the second spectrum of a module(II), Bull. Malays. Math. Sci. Soc. 39 (2016) (3), 1089–1103. Google Scholar

[5] S. Ballal and V. Kharat, Zariski topology on lattice modules, Asian-Eur. J. Math. 8 (2015) (4), 1550066 (10 pp). Google Scholar

[6] V. Borkar, P. Girase and N. Phadatare, Classical Zariski topology on prime spectrum of lattice modules, Journal of Algebra and Related Topics 6 (2018) (2), 1–14. Google Scholar

[7] V. Borkar, P. Girase and N. Phadatare, Zariski second radical elements of lattice modules, Asian-Eur. J. Math., doi:10.1142/S1793557121500558. Google Scholar

[8] N. Bourbaki, Algebre Commutative, Chap 1-2, Hermann, Paris, 1961. Google Scholar

[9] N. Bourbaki, Elements of Mathematics, General topology, Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966. Google Scholar

[10] F. Callialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-Module M, Turkish J. Math. 41 (2017) (2), 326–336. Google Scholar

[11] P. Girase, V. Borkar and N. Phadatare, On the classical prime spectrum of lattice modules, Int. Elect. J. Algebra 25 (2019), 186–198. Google Scholar

[12] P. Girase, V. Borkar and N. Phadatare, Zariski prime radical elements of lattice modules, Southeast Asian Bull. Math. 44 (2020) (3), 335–344. Google Scholar

[13] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. Google Scholar

[14] J. R. Munkres, Topology: a First Course, Prentice-Hall, Inc. Eglewood Cliffs, New Jersey, 1975. Google Scholar

[15] N. Phadatare, S. Ballal and V. Kharat, On the second spectrum of lattice modules, Discuss. Math. Gen. Algebra and Appl. 37 (2017) (1), 59–74. Google Scholar

[16] N. Phadatare and V. Kharat, On the second radical elements of lattice modules, Tbilisi Math. J. 11 (2018) (4), 165–173. Google Scholar

[17] N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, Algebra and its applications(New Delhi, 1981), Lecture Notes in Pure and Appl. Math., 91, Dekker, New York, (1984), 256–276. Google Scholar

[18] N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II:minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. (Szeged) 52 (1988) (1-2), 53–67. Google Scholar

[19] S. Yassemi, The dual notion of prime submodules, Arch. Math. (Brno), 37 (2001), 273–278. Google Scholar