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About the spectrum of Nagata rings

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Abstract

In this paper, we deal with the Nagata ring R(n) in case R is obtained by a (T, I, D) construction. We characterize when R(n) is a strong S-domain and catenarian. This study allows us to provide several interesting applications and examples.

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References

  1. Ameziane S., Elkhayari Z., Yengui I.: About the spectrum of the rings R(n) and R〈n〉. Commun. Algebra 31(3), 1047–1058 (2003)

    Article  MATH  Google Scholar 

  2. Anderson D.F., Bouvier A., Dobbs D.E., Fontana M., Kabbaj S.: On Jaffard domains. Expo. Math. 5, 145–175 (1988)

    MathSciNet  Google Scholar 

  3. Anderson D.F., Dobbs D.E., Kabbaj S., Mullay S.B.: Universally catenarian domains of D + M type. Proc. Am. Math. Soc. 104(2), 378–384 (1988)

    MATH  Google Scholar 

  4. Ayache A., Cahen P.J., Echi O.: Valuative heights and infinite Nagata rings. Commun. Algebra 23(5), 1913–1926 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ayache A., Cahen P.J., Echi O.: Anneaux quasi-prüfériens et P-anneaux. Boll. Un. Mat. Ital. B 10(7), 1–24 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Ayache A., Jarboui N.: On questions related to stably strong S-domains. J. Algebra 291, 164–170 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Nasr M., Jarboui N.: A counterexample for a conjecture about the catenarity of polynomial rings. J. Algebra 248, 785–789 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bouvier A., Dobbs D.E., Fontana M.: Catenarian integral domains. Adv. Math. 72, 211–238 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cahen P.J.: Couples d’anneaux partageant un idéal. Arch. Math. 51, 505–514 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cahen P.J.: Construction (B, I, D) et anneaux localement ou residuellement de Jaffard. Arch. Math. 54(2), 125–141 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cahen, P.J., Elkhayyari, Z., Kabbaj, S.: Krull and valuative dimension of the Serre conjecture ring R〈n〉. In: Lecture Notes in Pure and Applied Mathematics, vol. 185, pp. 173–180. Dekker, New York (1997)

  12. Fontana M., Izelgue L., Kabbaj S.: Sur quelques propriétés des sous-anneaux de la forme D + I d’un anneau intégre. Commun. Algebra 23(11), 4189–4210 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gilmer R.: Multiplicative Ideal Theory. Dekker, New York (1972)

    MATH  Google Scholar 

  14. Hedstrom J.R., Houston E.G.: Pseudo-valuation domains. Pac. J. Math. 75, 137–147 (1978)

    MathSciNet  MATH  Google Scholar 

  15. Jaffard, P.: Théorie de la dimension dans les anneaux de polynômes. Mem. Sc. Math, 146. Gauthier-Villars, Paris (1960)

  16. Jarboui N., Jerbi A.: Pullbacks and universal catenarity. Publ. Math. 52, 365–375 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Kabbaj S.: La formule de la dimension pour les S-domaines forts universels. Boll. Un. Mat. Ital. D 5(6), 145–161 (1986)

    MathSciNet  MATH  Google Scholar 

  18. Kabbaj S.: Sur les S-domaines forts de Kaplansky. J. Algebra 137(2), 400–414 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kaplansky I.: Commutative Rings (revised edition). University Press, Chicago (1974)

    Google Scholar 

  20. Malik S., Mott J.L.: Strong S-domains. J. Pure Appl. Algebra 28(3), 249–264 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nagata M.: Local Rings. Interscience, New York (1962)

    MATH  Google Scholar 

  22. Yengui I.: Two counterexamples about the Nagata and Serre conjecture rings. J. Pure Appl. Algebra 153(2), 191–195 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yengui I.: On questions related to saturated chains of primes in polynomial rings. J. Pure Appl. Algebra 178(2), 215–224 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, King Faisal University, P. O. Box 380, Hofuf, 31982, Saudi Arabia

    Mabrouk Ben Nasr, Noômen Jarboui & Nasr Zeyeda

Authors
  1. Mabrouk Ben Nasr
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  2. Noômen Jarboui
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  3. Nasr Zeyeda
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Corresponding author

Correspondence to Noômen Jarboui.

Additional information

Communicated by John S. Wilson.

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Nasr, M.B., Jarboui, N. & Zeyeda, N. About the spectrum of Nagata rings. Monatsh Math 167, 257–272 (2012). https://doi.org/10.1007/s00605-011-0325-1

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  • DOI: https://doi.org/10.1007/s00605-011-0325-1

Keywords

Mathematics Subject Classification (2000)