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

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In this paper, we deal with the study of intermediate domains between a domain and a domain such that is an intersection of localizations of , namely the pair . More precisely, we study the pair and the pair , where and . We prove that, if is a Jaffard domain, then is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if is an -domain, then is a residually algebraic pair (that is for each intermediate domain between and , if is a prime ideal of , then is algebraic over ). Moreover, the pair is if and only if is , for some properties . Lastly, we answer in the positive a question raised in [7] by D. F. Anderson and D. N. Elabidine: we show that if is a Jaffard local domain with maximal ideal , then the domain is a Jaffard domain.

How to cite

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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

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  1. ANDERSON, D. D.- ANDERSON, D. F.- ZAFRULLAH, M., Rings between and , 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 , 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 , , 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 -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

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