Intermediate domains between a domain and some intersection of its localizations

Mabrouk Ben Nasr; Noômen Jarboui

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 3, page 701-713
  • ISSN: 0392-4041

Abstract

top
In this paper, we deal with the study of intermediate domains between a domain R and a domain T such that T is an intersection of localizations of R , namely the pair R , T . More precisely, we study the pair R , R d and the pair R , R ~ , where R d = ∩ R M ∣ M ∈ Max ⁢ R , h ⁢ t ⁢ M = dim ⁡ R and R ~ = ∩ R M ∣ M ∈ Max ⁢ R , h ⁢ t ⁢ M ≥ 2 . We prove that, if R is a Jaffard domain, then R , R d ⁢ n is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if R is an S -domain, then R , R ~ is a residually algebraic pair (that is for each intermediate domain S between R and R ~ , if Q is a prime ideal of S , then S / Q is algebraic over R / Q ∩ R ). Moreover, the pair R , R ~ is P if and only if R is P , for some properties P . Lastly, we answer in the positive a question raised in [7] by D. F. Anderson and D. N. Elabidine: we show that if R is a Jaffard local domain with maximal ideal M , then the domain R ♯ = ∩ R p ∣ p ⊂ M is a Jaffard domain.

How to cite

top

Ben Nasr, Mabrouk, and Jarboui, Noômen. "Intermediate domains between a domain and some intersection of its localizations." Bollettino dell'Unione Matematica Italiana 5-B.3 (2002): 701-713. <http://eudml.org/doc/195772>.

@article{BenNasr2002,
abstract = {In this paper, we deal with the study of intermediate domains between a domain $R$ and a domain $T$ such that $T$ is an intersection of localizations of $R$, namely the pair $(R, T)$. More precisely, we study the pair $(R, R_\{d\})$ and the pair $(R,\tilde\{R\})$, where $R_\{d\}=\cap\\{R_\{M\} \mid M \in \text\{Max\}(R), htM = \dim R \\}$ and $\tilde\{R\}= \cap \\{R_\{M\} \mid M\in \text\{Max\}(R), htM \geq 2 \\}$. We prove that, if $R$ is a Jaffard domain, then $(R, R_\{d\}[n])$ is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if $R$ is an $S$-domain, then $(R,\tilde\{R\})$ is a residually algebraic pair (that is for each intermediate domain $S$ between $R$ and $\tilde\{R\}$, if $Q$ is a prime ideal of $S$ , then $S/Q$ is algebraic over $R/(Q\cap R)$). Moreover, the pair $(R,\tilde\{R\})$ is $\mathcal\{P\}$ if and only if $R$ is $\mathcal\{P\}$, for some properties $\mathcal\{P\}$. Lastly, we answer in the positive a question raised in [7] by D. F. Anderson and D. N. Elabidine: we show that if $R$ is a Jaffard local domain with maximal ideal $M$, then the domain $R^\{\sharp\} =\cap\\{R_\{p\} \mid p \subset M\\}$ is a Jaffard domain.},
author = {Ben Nasr, Mabrouk, Jarboui, Noômen},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {701-713},
publisher = {Unione Matematica Italiana},
title = {Intermediate domains between a domain and some intersection of its localizations},
url = {http://eudml.org/doc/195772},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Ben Nasr, Mabrouk
AU - Jarboui, Noômen
TI - Intermediate domains between a domain and some intersection of its localizations
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/10//
PB - Unione Matematica Italiana
VL - 5-B
IS - 3
SP - 701
EP - 713
AB - In this paper, we deal with the study of intermediate domains between a domain $R$ and a domain $T$ such that $T$ is an intersection of localizations of $R$, namely the pair $(R, T)$. More precisely, we study the pair $(R, R_{d})$ and the pair $(R,\tilde{R})$, where $R_{d}=\cap\{R_{M} \mid M \in \text{Max}(R), htM = \dim R \}$ and $\tilde{R}= \cap \{R_{M} \mid M\in \text{Max}(R), htM \geq 2 \}$. We prove that, if $R$ is a Jaffard domain, then $(R, R_{d}[n])$ is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if $R$ is an $S$-domain, then $(R,\tilde{R})$ is a residually algebraic pair (that is for each intermediate domain $S$ between $R$ and $\tilde{R}$, if $Q$ is a prime ideal of $S$ , then $S/Q$ is algebraic over $R/(Q\cap R)$). Moreover, the pair $(R,\tilde{R})$ is $\mathcal{P}$ if and only if $R$ is $\mathcal{P}$, for some properties $\mathcal{P}$. Lastly, we answer in the positive a question raised in [7] by D. F. Anderson and D. N. Elabidine: we show that if $R$ is a Jaffard local domain with maximal ideal $M$, then the domain $R^{\sharp} =\cap\{R_{p} \mid p \subset M\}$ is a Jaffard domain.
LA - eng
UR - http://eudml.org/doc/195772
ER -

References

top
  1. ANDERSON, D. D.- ANDERSON, D. F.- ZAFRULLAH, M., Rings between D ⁢ X and K ⁢ X , Houston, J. Math., 17 (1991), 109-129. Zbl0736.13015MR1107192
  2. ANDERSON, D. F.- BOUVIER, A., Ideal transforms and overrings of a quasilocal integral domain, Ann. Univ. Ferrara, Sez. VII, Sc. Math., 32 (1986), 15-38. Zbl0655.13002MR901583
  3. ANDERSON, D. F.- BOUVIER, A.- DOBBS, D. E.- FONTANA, M.- KABBAJ, S., On Jaffard domains, Expo. Math., 5 (1988), 145-175. Zbl0657.13011MR938180
  4. AYACHE, A.- CAHEN, P.-J., Anneaux vérifiant absolument l'inégalité ou la formule de la dimension, Boll. Un. Math. Ital. B(7)6, n-1 (1992), 39-65. Zbl0785.13001MR1164937
  5. AYACHE, A.- CAHEN, P.-J., Radical valuatif et sous-extensions, Comm. Algebra., 26 (9) (1998), 2767-2787. Zbl0933.13008MR1635917
  6. AYACHE, A.- JABALLAH, A., Residually algebraic pairs of rings, Math. Z., 225 (1997), 49-65. Zbl0868.13007MR1451331
  7. ANDERSON, D. F.- NOUR EL ABIDINE, D., Some remarks on the ring R ♯ , Lect. Notes. Pure. App. Math.M. Dekker, New York, 185 (1997), 33-44. Zbl0896.13009MR1422464
  8. BEN NASR, M.- ECHI, O.- IZELGUE, L.- JARBOUI, N., Pairs of domains where all intermediate domains are Jaffard, J. Pure. Appl. Algebra, 145 (2000), 1-18. Zbl1079.13510MR1732284
  9. CAHEN, P.-J., Couples d'anneaux partageant un idéal, Arch. Math., 51 (1988), 505-514. Zbl0668.13005MR973725
  10. CAHEN, J., Construction B , I , D et anneaux localement ou residuellement de Jaffard, Arch. Math, vol. 54 (1990), 125-141. Zbl0707.13004MR1035345
  11. DAVIS, E., Integrally closed pairs, Conf. comm. Alg.Lec. Notes. Math, vol. 311, Springer-Verlag, Berlin and New York, (1973), 103-106. Zbl0248.13005MR335490
  12. DOBBS, E.- FONTANA, M., Universally incomparable ring homomorphisms, Bull. Austral. Math. Soc., 29 (1984), 289-302. Zbl0535.13006MR748722
  13. ECHI, O., Sur les hauteurs valuatives, Boll. Un. Mat. Ital. (7)9-B (1995), 281-297. Zbl0849.13002MR1333963
  14. FONTANA, M., Topologically defined classes of commutative rings, Ann. Math. Pura. Appl., 123 (1980), 331-355. Zbl0443.13001MR581935
  15. GILMER, R., Multiplicative ideal theory, Marcel Dekker, New-York (1972). Zbl0248.13001MR427289
  16. KAPLANSKY, I., Commutative rings, The University of Chicago press (Revised edition) (1974). Zbl0296.13001MR345945
  17. MALIK, S.- MOTT, J. L., Strong S -domains, J. Pure. Appl. Alg., 28 (1983), 249-264. Zbl0536.13001MR701353
  18. NAGATA, M., A general theory of algebraic geometry over Dedekind domains I, Am. J. Math., 78 (1956), 78-116. Zbl0089.26403MR82725
  19. NAGATA, M., Local rings, Interscience Tracts in Pure. Appl. Math, no. 13, Interscience, New York, (1962). MR 27 ll--5790. Zbl0123.03402MR155856
  20. WADSWORTH, A. R., Pairs of domains where all intermediate domains are Noetherian, Tran. Amer. Math. Soc., 195 (1974), 201-211. Zbl0294.13010MR349665

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.