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A Zariski Topology for Bicomodules and Corings

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Abstract

In this paper we introduce and investigate top (bi)comodules of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative) rings. We restrict our attention in this paper to duo (bi)comodules (satisfying suitable conditions) and study the interplay between the coalgebraic properties of such (bi)comodules and the introduced Zariski topology. In particular, we apply our results to introduce a Zariski topology on the fully coprime spectrum of a given non-zero coring considered canonically as duo object in its category of bicomodules.

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References

  1. Abuhlail, J.Y.: Fully coprime comodules and fully coprime corings. Appl. Categ. Structures 14(5–6), 379–409 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abuhlail, J.Y.: Rational modules for corings. Comm. Algebra 31, 5793–5840 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Atiyah M., Macdonald, I.: Introduction to Commutative Algebra. Addison-Wesley, Reading, MA (1969)

    MATH  Google Scholar 

  4. Annin, S.: Associated and attached primes over noncommutative rings. Ph.D. Dissertation, University of California at Berkeley (2002)

  5. Bican, L., Jambor, P., Kepka, T., Nĕmec, P.: Prime and coprime modules. Fund. Math. 107(1), 33–45 (1980)

    MATH  MathSciNet  Google Scholar 

  6. Bourbaki, N.: Commutative Algebra. Springer, Berlin (1998)

    Google Scholar 

  7. Bourbaki, N. General Topology, Part I. Addison-Wesley, Reading, MA (1966)

    Google Scholar 

  8. Brzezinski, T., Wisbauer, R.: Corings and comodules. London Mathematical Society Lecture Note Series, vol. 309. Cambridge University Press, Cambridge, MA (2003)

    Google Scholar 

  9. Faith, C.: Algebra II, Ring Theory. Springer, Berlin (1976)

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  11. McCasland, R., Moore, M., Smith, P.: On the spectrum of a module over a commutative ring. Comm. Algebra 25, 79–103 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Nekooei, R., Torkzadeh, L.: Topology on coalgebras. Bull. Iranian Math. Soc. 27(2), 45–63 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Raggi, F., Ríos Montes, J., Wisbauer, R.: Coprime preradicals and modules. J. Pure Appl. Algebra 200, 51–69 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wijayanti, I.: Coprime modules and comodules. Ph.D. Dissertation, Heinrich-Heine Universität, Düsseldorf (2006)

    Google Scholar 

  15. Wisbauer, R.: Foundations of Module and Ring Theory. A Handbook for Study and Research. Gordon and Breach, New York (1991)

    MATH  Google Scholar 

  16. Zimmermann-Huisgen, B.: Pure submodules of direct products of free modules. Math. Ann. 224, 233–245 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang, G.: Spectra of modules over any ring. Nanjing Univ. J. Math. Biquarterly 16(1), 42–52 (1999)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Box # 5046, 31261, Dhahran, Saudi Arabia

    Jawad Y. Abuhlail

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  1. Jawad Y. Abuhlail
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Correspondence to Jawad Y. Abuhlail.

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Supported by King Fahd University of Petroleum & Minerals, Research Project # INT/296.

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Abuhlail, J.Y. A Zariski Topology for Bicomodules and Corings. Appl Categor Struct 16, 13–28 (2008). https://doi.org/10.1007/s10485-007-9088-1

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  • DOI: https://doi.org/10.1007/s10485-007-9088-1

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