Some remarks on the altitude inequality

Noômen Jarboui

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 1, page 39-52
  • ISSN: 0010-1354

Abstract

top
We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization of integral extensions.

How to cite

top

Jarboui, Noômen. "Some remarks on the altitude inequality." Colloquium Mathematicae 80.1 (1999): 39-52. <http://eudml.org/doc/210704>.

@article{Jarboui1999,
abstract = {We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization of integral extensions.},
author = {Jarboui, Noômen},
journal = {Colloquium Mathematicae},
keywords = {valuation domain; altitude inequality; restrictive altitude inequality; locally Jaffard domain; heights; altitude formula; locally Jaffard domains; characterization of integral extensions},
language = {eng},
number = {1},
pages = {39-52},
title = {Some remarks on the altitude inequality},
url = {http://eudml.org/doc/210704},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Jarboui, Noômen
TI - Some remarks on the altitude inequality
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 1
SP - 39
EP - 52
AB - We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization of integral extensions.
LA - eng
KW - valuation domain; altitude inequality; restrictive altitude inequality; locally Jaffard domain; heights; altitude formula; locally Jaffard domains; characterization of integral extensions
UR - http://eudml.org/doc/210704
ER -

References

top
  1. [1] D. F. Anderson, A. Bouvier, D. E. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domains, Exposition. Math. 6 (1988), 145-175. Zbl0657.13011
  2. [2] A. Ayache et P.-J. Cahen, Anneaux vérifiant absolument l'inégalité ou la formule de la dimension, Boll. Un. Mat. Ital. B (7) 6 (1992), 39-65. 
  3. [3] A. Ayache, P.-J. Cahen et O. Echi, Anneaux quasi-prüfériens et P-anneaux, ibid. 10 (1996), 1-24. 
  4. [4] A. Ayache and A. Jaballah, Residually algebraic pairs of rings, Math. Z. 225 (1997), 49-65. 
  5. [5] M. Bennasr, O. Echi, L. Izelgue and N. Jarboui, Pairs of domains where all intermediate domains are Jaffard, J. Pure Appl. Algebra, to appear. Zbl1079.13510
  6. [5*] M. Bennasr and N. Jarboui, Intermediate domains between a domain and some intersection of its localizations, submitted for publication. 
  7. [6] N. Bourbaki, Algèbre commutative, Chapitres 5 et 6, Masson, Paris, 1985. 
  8. [7] A. Bouvier, D. E. Dobbs and M. Fontana, Universally catenarian integral domains, Adv. Math. 72 (1988), 211-238. 
  9. [8] P.-J. Cahen, Couples d'anneaux partageant un idéal, Arch. Math. (Basel) 51 (1988), 505-514. Zbl0668.13005
  10. [9] P.-J. Cahen, Construction B, I, D et anneaux localement ou résiduellement de Jaffard, ibid. 54 (1990), 125-141. 
  11. [10] D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), 353-363. Zbl0326.13002
  12. [11] D. E. Dobbs, Lying over pairs of commutative rings, Canad. J. Math. 33 (1987), 454-475. Zbl0466.13002
  13. [12] O. Echi, Sur les hauteurs valuatives, Boll. Un. Mat. Ital. B (7) 9 (1995), 281-297. 
  14. [13] M. Fontana, L. Izelgue et S. Kabbaj, Quelques propriétés des chaînes d’idéaux premiers dans les anneaux A + X B [ X ] , Comm. Algebra 22 (1994), 9-24. 
  15. [14] R. Gilmer, Mulltiplicative Ideal Theory, Marcel Dekker, New York, 1972. 
  16. [15] P. Jaffard, Théorie de la dimension dans les anneaux de polynômes, Mém. Sci. Math. 146, Gauthier-Villars, Paris, 1960. Zbl0096.02502
  17. [16] S. Kabbaj, La formule de la dimension pour les S-domaines forts universels, Boll. Un. Mat. Ital. (6) 5 (1986), 145-161. Zbl0651.13008
  18. [17] I. Kaplansky, Commutative Rings, rev. ed., Univ. of Chicago Press, Chicago 1974. Zbl0296.13001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.