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SECOND CLASSICAL ZARISKI TOPOLOGY ON SECOND SPECTRUM OF LATTICE MODULES

  • Received : 2019.12.25
  • Accepted : 2020.07.21
  • Published : 2020.09.30

Abstract

Let M be a lattice module over a C-lattice L. Let Specs(M) be the collection of all second elements of M. In this paper, we consider a topology on Specs(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 Specs(M). We investigate this topological space from the point of view of spectral spaces. We show that Specs(M) is always T0-space and each finite irreducible closed subset of Specs(M) has a generic point.

Keywords

References

  1. F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commutative ideal theory, Period. Math. Hungar. 30 (1995) (1), 1-26. https://doi.org/10.1007/BF01876923
  2. E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Damascus Univ. Basic Sci. 19 (2003) (2), 9-21.
  3. H. Ansari-Toroghy and F. Farshadifar, The Zariski topology on the second spectrum of a module, Algebra Colloq. 21 (2014) (4), 671-688. https://doi.org/10.1142/S1005386714000625
  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. https://doi.org/10.1007/s40840-015-0225-y
  5. S. Ballal and V. Kharat, Zariski topology on lattice modules, Asian-Eur. J. Math. 8 (2015) (4), 1550066 (10 pp). https://doi.org/10.1142/S1793557115500667
  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.
  7. V. Borkar, P. Girase and N. Phadatare, Zariski second radical elements of lattice modules, Asian-Eur. J. Math., doi:10.1142/S1793557121500558.
  8. N. Bourbaki, Algebre Commutative, Chap 1-2, Hermann, Paris, 1961.
  9. N. Bourbaki, Elements of Mathematics, General topology, Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966.
  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. https://doi.org/10.3906/mat-1502-31
  11. P. Girase, V. Borkar and N. Phadatare, On the classical prime spectrum of lattice modules, Int. Elect. J. Algebra 25 (2019), 186-198.
  12. P. Girase, V. Borkar and N. Phadatare, Zariski prime radical elements of lattice modules, Southeast Asian Bull. Math. 44 (2020) (3), 335-344.
  13. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. https://doi.org/10.1090/S0002-9947-1969-0251026-X
  14. J. R. Munkres, Topology: a First Course, Prentice-Hall, Inc. Eglewood Cliffs, New Jersey, 1975.
  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. https://doi.org/10.7151/dmgaa.1266
  16. N. Phadatare and V. Kharat, On the second radical elements of lattice modules, Tbilisi Math. J. 11 (2018) (4), 165-173. https://doi.org/10.32513/tbilisi/1546570892
  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.
  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.
  19. S. Yassemi, The dual notion of prime submodules, Arch. Math. (Brno), 37 (2001), 273-278.