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The catenarian property of power series rings over a globalized pseudo-valuation domain

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Abstract

LetR be a locally finite dimensional globalized pseudo-valuation domain with Arnold’s SFT-property. It is shown that the power series ringR[[X]] is catenarian. Examples of non catenarianR[[X]] (resp.R[X]) withR[X] (resp.R[[X]]) catenarian are also given.

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References

  1. Anderson D.F., Dobbs D.E.,Pairs of rings with the same prime ideals, Can. J. Math.,32 (1980), 362–384.

    MATH  MathSciNet  Google Scholar 

  2. Arnold J.T.,Krull dimension in power series rings, Trans. Amer. Math. Soc.,177 (1973), 299–304.

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold J.T.,Power series rings over Prüfer domains, Pac. J. Math.,44 (1973), 1–11.

    MATH  Google Scholar 

  4. Arnold J.T.,The catenarian property of power series rings over a Prüfer domain, Proc. Amer. Math. Soc.,94 (1985), 577–580.

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourbaki N.,Algèbre commutative, Chap. 8–9, Masson, Paris, 1983.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Bouvier A., Fontana M.,The catenarian property of the polynomial rings over a Prüfer domain, in Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math, n. 1146, Springer, Berlin-New York, 1985.

    Google Scholar 

  8. Brewer J.W.,Power series over commutative rings, M. Dekker, New York, 1981.

    MATH  Google Scholar 

  9. Dobbs D.E., Fontana M.,On pseudo-valuation domains and their globalizations, Commutative Algebra, Proc. Trento Conference, Lect. Notes Pure Appl. Math., M. Dekker, 1983.

  10. Dobbs D.E., Fontana M.,Locally pseudo-valuation domains, Ann. Mat. Pura Appl.,134 (1983), 147–168.

    Article  MATH  MathSciNet  Google Scholar 

  11. Dugundji J.,Topology, Allyn and Bacon, Boston, 1969.

    Google Scholar 

  12. Fontana M.,Topologically defined classes of commutative rings, Ann. Mat. Pura Appl.,123 (1980), 331–355.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  14. Girolami F.,Power series rings over globalized pseudo-valuation domains, J. Pure Appl. Algebra,50 (1988), 259–269.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Hedstrom J.R., Houston E.G.,Pseudo-valuation domains, II, Houston J. Math.,4 (1978), 199–207.

    MATH  MathSciNet  Google Scholar 

  17. Lequain Y.,Catenarian property in a domain of formal power series, J. Algebra,65 (1980), 110–117.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Ratliff L.J., Jr.,On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals, (I), Amer. J. Math.,91 (1969), 508–528.

    Article  MATH  MathSciNet  Google Scholar 

  20. Ratliff L.J., Jr.,Chain Conjectures in Ring Theory, Lecture Notes in Math. n. 647, Springer, Berlin-New York, 1978.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Istituto G. Castelnuovo, Università di Roma “La Sapienza”, Piazzale A. Moro, 5-00185, Roma, Italia

    Florida Girolami

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  1. Florida Girolami
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Supported by 60% MPI Research Fund.

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Girolami, F. The catenarian property of power series rings over a globalized pseudo-valuation domain. Rend. Circ. Mat. Palermo 38, 5–12 (1989). https://doi.org/10.1007/BF02844845

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  • DOI: https://doi.org/10.1007/BF02844845

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