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The Zariski Topology on the Second Spectrum of a Module (II)

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Abstract

Let R be a commutative ring and M an R-module. Let \(Spec^s(M)\) be the collection of all second submodules of M. In this article, we consider a new topology on \(Spec^s(M)\), called the second classical Zariski topology, and investigate the interplay between the module theoretic properties of M and the topological properties of \(Spec^s(M)\). Moreover, we study \(Spec^s(M)\) from point of view of spectral space.

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References

  1. Abuhlail, J.Y.: Zariski Topologies for Coprime and Second Submodules, to appear in Algebra Colloquium

  2. Abuhlail, J.Y.: A dual Zariski topology for modules. Topol. Appl. 158(3), 457–467 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ansari-Toroghy, H., Farshadifar, F.: The dual notion of multiplication modules. Taiwan. J. Math. 11(4), 1189–1201 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Ansari-Toroghy, H., Farshadifar, F.: The dual notions of some generalizations of prime submodules. Commun. Algebra 39(7), 2396–2416 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ansari-Toroghy, H., Farshadifar, F.: On the dual notion of prime submodules. Algebra Colloq. 19(Spec 1), 1109–1116 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ansari-Toroghy, H., Farshadifar, F.: On the dual notion of prime submodules (II). Mediterr. J. Math. 9(2), 329–338 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ansari-Toroghy, H., Farshadifar, F.: The Zariski Topology on the Second Spectrum of a Module. Algebra Colloq. 21(4), 671–688 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ansari-Toroghy, H., Farshadifar, F.: On the dual notion of prime radicals of submodules. Asian Eur. J. Math. 6(2), 1350024 (2013). (11 pages)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ansari-Toroghy, H., Farshadifar, F.: On the dual notion of multiplication modules. AJSE-Math. 36(6), 925–932 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Ansari-Toroghy, H., Keyvani, S.: Strongly top modules. Bull. Malays. Math. Sci. Soc. 37(1), 73–82 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Ansari-Toroghy, H., Keyvani, S.: On the maximal spectrum of a module and Zariski topology. Bull. Malays. Math. Sci. Soc. 38(2), 303–316 (2015)

  12. Behboodi, M., Haddadi, M.R.: Classical Zariski topology of modules and spectral spaces I. Int. Electron. J. Algebra 4, 104–130 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Behboodi, M., Haddadi, M.R.: Classical Zariski topology of modules and spectral spaces II. Int. Electron. J. Algebra 4, 131–148 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Ceken, S., Alkan, M., Smith, P.F.: Second modules over noncommutative rings. Commun. Algebra 41(1), 83–98 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ceken, S., Alkan, M., Smith, P.F.: The dual notion of the prime radical of a module. J. Algebra 392, 255–265 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hochster, M.: Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hochster, M.: The minimal prime spectrum of a commutative ring. Can. J. Math. 23, 749–758 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, H.C., Shum, K.P.: On a problem of spectral posets. J. Appl. Algebra Discret. Struct. 1(3), 203–209 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Lu, C.P.: The Zariski topology on the prime spectrum of a module. Houst. J. Math. 25(3), 417–432 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Lu, C.P.: Modules with Noetherian spectrum. Commun. Algebra 38, 807–828 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Munkres, J.R.: Topology. A First Course. Prentice Hall, Upper Saddle River (1975)

    MATH  Google Scholar 

  22. Vamos, P.: The dual of the notion of “finitely generated”. J. Lond. Math. Soc 43, 643–646 (1965)

    MathSciNet  MATH  Google Scholar 

  23. Van Sanh, N., Thas, L.P., Al-Mepahi, N.F.A., Shum, K.P.: Zariski topology of prime spectrum of a module. In: Proceedings of the International conference 2010 held in Gajha Mada Univesity, Novenber, 2011, Yojakarta, Indonesia (Abvance in algebraic structure). World Scientific, Inc., pp. 461-477 (2011)

  24. Yassemi, S.: The dual notion of prime submodules. Arch. Math (Brno) 37, 273–278 (2001)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank Dr. R. Ovlyaee-Sarmazdeh for his helpful suggestions. Also we are grateful to the referee for careful reading of our manuscript and useful comments.

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Authors and Affiliations

  1. Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, P. O. Box 41335-19141, Rasht, Iran

    H. Ansari-Toroghy & S. Keyvani

  2. University of Farhangian, Tehran, Iran

    F. Farshadifar

Authors
  1. H. Ansari-Toroghy
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  2. S. Keyvani
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  3. F. Farshadifar
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Corresponding author

Correspondence to H. Ansari-Toroghy.

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Communicated by Siamak Yassemi.

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Ansari-Toroghy, H., Keyvani, S. & Farshadifar, F. The Zariski Topology on the Second Spectrum of a Module (II). Bull. Malays. Math. Sci. Soc. 39, 1089–1103 (2016). https://doi.org/10.1007/s40840-015-0225-y

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  • DOI: https://doi.org/10.1007/s40840-015-0225-y

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