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A note on maximal non-Noetherian subrings of a domain

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Abstract

This paper is concerned with the prime spectrum of maximal non-Noetherian subrings of a given domain. It is proved that if R is a maximal non-Noetherian subring of S, then R is a stably strong S-domain and that R is universally catenarian iff S is universally catenarian. Our main results lead to new examples of stably strong S-domains and universally catenarian domains. The relationship with n-dimensional pairs and residually Mori pairs is established.

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References

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

    MathSciNet  Google Scholar 

  • 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 

  • Ayache A., Ben Nasr M., Echi O., Jarboui N.: Universally catenarian and going-down pairs of rings. Math. Z. 238, 695–731 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Ayache, A., Cahen, P.-J.: Anneaux vérifiant absolument l’inégalité ou la formule de la dimension. Boll. Un. Mat. Ital. B (7) 6(1), 39–65 (1992)

    Google Scholar 

  • Ayache A., Cahen P.-J., Echi O.: Intersections de produits fibrés et formule de la dimension. Commun. Algebra 22(9), 3495–3504 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Ayache A., Dobbs D.E., Echi O.: On maximal non ACCP subrings. J. Algebra Appl. 6(5), 873–894 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Ayache A., Jaballah A.: Residually algebraic pairs of rings. Math. Z. 225, 49–65 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Ayache A., Jarboui N.: Maximal non-Noetherian subrings of a domain. J. Algebra 248, 806–823 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Ayache A., Jarboui N.: Universally catenarian domains of the type A + I. Ric. Mat. 57, 27–42 (2008)

    Article  MathSciNet  Google Scholar 

  • Barucci, V.: Mori domains. In: Non Noetherian Commutative Ring Theory. Mathematics and its Applications, vol. 520, pp. 57–73. Kluwer, Dordrecht (2000)

  • Ben Abdallah M.J., Jarboui N.: On universally catenarian pairs. J. Pure Appl. Algebra 212(10), 2170–2175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Ben Nasr M., Jarboui N.: Maximal non-Jaffard subring of a field. Publ. Math. 4, 157–175 (2000)

    MathSciNet  Google Scholar 

  • Ben Nasr, M., Jarboui, N.: Intermediate domains between a domain and some intersection of its localizations. Boll. Un. Mat. Ital. (8) 5(3), 701–713 (2002)

    Google Scholar 

  • 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chatham R.D., Dobbs D.E.: Pairs of commutative rings in which all intermediate rings have the same dimension. Houst. J. Math. 33(3), 635–647 (2007)

    MathSciNet  MATH  Google Scholar 

  • Gilbert, M.S.: Extensions of commutative rings with linearly ordered intermediate rings. Ph.D. dissertation, University of Tennessee, Knoxville (1996)

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

    MATH  Google Scholar 

  • Gilmer R., Hoffmann J.: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60(1), 81–85 (1975)

    MathSciNet  MATH  Google Scholar 

  • Jaffard, P.: Théorie de la Dimension dans les Anneaux de Polynômes. Mém. Sc. Math., vol. 146. Gauthier-Villars, Paris (1960)

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

    MathSciNet  MATH  Google Scholar 

  • Kabbaj, S.: La formule de la dimension pour les S-domaines forts universels. Boll. Un. Mat. Ital. D (6) 5(1), 145–161 (1986)

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

    Article  MathSciNet  MATH  Google Scholar 

  • Kaplansky I.: Commutative Rings (revised edn). The University of Chicago Press, Chicago (1974)

    Google Scholar 

  • Lucas, T.: Examples built in the D + M, A + XB[X], and other pullback-constructions. In: Chapman, S., Glaz, S. (eds.) Non-Noetherian commutative Ring Theory, pp. 341–368. Kluwer, Norwell (2000)

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

    Article  MathSciNet  MATH  Google Scholar 

  • Visweswaran S.: Intermediate rings between D + I and K[y 1,…, y t ]. Commun. Algebra 18(2), 309–345 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Visweswaran S.: When is (K + I, K[y 1,…, y t ]) a Mori pair. Commun. Algebra 32(8), 3095–3110 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Wadsworth A.R.: Pairs of domains where all intermediate domains are Noetherian. Trans. Am. Math. Soc. 195, 201–211 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Sfax, P.O. Box 1171, 3000, Sfax, Tunisia

    Noômen Jarboui & Ayada Jerbi

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  1. Noômen Jarboui
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  2. Ayada Jerbi
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Correspondence to Noômen Jarboui.

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Jarboui, N., Jerbi, A. A note on maximal non-Noetherian subrings of a domain. Beitr Algebra Geom 53, 159–172 (2012). https://doi.org/10.1007/s13366-011-0055-5

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