Intermediate domains between a domain and some intersection of its localizations

Ben Nasr, Mabrouk ; Jarboui, Noômen

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 701-713.

Abstract (VO)

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.

Riassunto

In questo lavoro vengono studiati gli anelli compresi tra un dominio integro $R$ ed un suo sopranello $T$, definito tramite una intersezione di localizzazioni di $R$. In particolare, vengono studiate le coppie $(R, R_{d})$ ed $(R,\tilde{R})$ dove $R_{d}=\cap\{R_{M} \mid M\in \text{Max}(R), htM = \dim R \}$ ed $\tilde{R}= \cap \{R_{M} \mid M \in \text{Max}(R), htM \geq 2 \}$. Si dimostra che, se $R$ è un dominio di Jaffard, allora $(R, R_{d}[n])$ è una coppia di Jaffard; tale risultato generalizza [5, Théorème 1.9]. Si dimostra anche che, se $R$ è un $S$-dominio, allora $(R, \tilde{R})$ è una coppia residualmente algebrica (i.e. per ogni dominio intermedio $S$ tra $R$ e $\tilde{R}$ e per ogni ideale primo $Q$ di $S$, il dominio quoziente $S/Q$ è algebrico su $R / (Q\cap R)$). Inoltre, la coppia $(R,\tilde{R})$ è $\mathcal{P}$ se e soltanto se $R$ è $\mathcal{P}$, per una qualche proprietà $\mathcal{P}$. Infine, viene data una risposta affermativa ad una questione sollevata in [7] da D. F. Anderson e D. N. Elabidine: se $R$ è un dominio locale di Jaffard con ideale massimale $M$, allora il dominio $R^{\sharp} =\cap\{R_{p} \mid p \subset M\}$ è un dominio di Jaffard.

Bibliographie

[1] D. D. Anderson-D. F. Anderson-M. Zafrullah, Rings between $D[X]$ and $K[X]$, Houston, J. Math., 17 (1991), 109-129. | MR | Zbl

[2] D. F. Anderson-A. Bouvier, Ideal transforms and overrings of a quasilocal integral domain, Ann. Univ. Ferrara, Sez. VII, Sc. Math., 32 (1986), 15-38. | MR | Zbl

[3] D. F. Anderson-A. Bouvier-D. E. Dobbs-M. Fontana-S. Kabbaj, On Jaffard domains, Expo. Math., 5 (1988), 145-175. | MR | Zbl

[4] A. Ayache-P.-J. Cahen, Anneaux vérifiant absolument l'inégalité ou la formule de la dimension, Boll. Un. Math. Ital. B(7) 6, n-1 (1992), 39-65. | MR | Zbl

[5] A. Ayache-P.-J. Cahen, Radical valuatif et sous-extensions, Comm. Algebra., 26 (9) (1998), 2767-2787. | MR | Zbl

[6] A. Ayache-A. Jaballah, Residually algebraic pairs of rings, Math. Z., 225 (1997), 49-65. | MR | Zbl

[7] D. F. Anderson-D. Nour El Abidine, Some remarks on the ring $\mathcal{R}^{\sharp}$, Lect. Notes. Pure. App. Math. M. Dekker, New York, 185 (1997), 33-44. | MR | Zbl

[8] M. Ben Nasr-O. Echi-L. Izelgue-N. Jarboui, Pairs of domains where all intermediate domains are Jaffard, J. Pure. Appl. Algebra, 145 (2000), 1-18. | MR | Zbl

[9] P.-J. Cahen, Couples d'anneaux partageant un idéal, Arch. Math., 51 (1988), 505-514. | MR | Zbl

[10] J. Cahen, Construction $B$, $I$, $D$ et anneaux localement ou residuellement de Jaffard, Arch. Math, vol. 54 (1990), 125-141. | MR | Zbl

[11] E. Davis, Integrally closed pairs, Conf. comm. Alg. Lec. Notes. Math, vol. 311, Springer-Verlag, Berlin and New York, (1973), 103-106. | MR | Zbl

[12] E. Dobbs-M. Fontana, Universally incomparable ring homomorphisms, Bull. Austral. Math. Soc., 29 (1984), 289-302. | MR | Zbl

[13] O. Echi, Sur les hauteurs valuatives, Boll. Un. Mat. Ital. (7) 9-B (1995), 281-297. | MR | Zbl

[14] M. Fontana, Topologically defined classes of commutative rings, Ann. Math. Pura. Appl., 123 (1980), 331-355. | MR | Zbl

[15] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New-York (1972). | MR | Zbl

[16] I. Kaplansky, Commutative rings, The University of Chicago press (Revised edition) (1974). | MR | Zbl

[17] S. Malik-J. L. Mott, Strong $S$-domains, J. Pure. Appl. Alg., 28 (1983), 249-264. | MR | Zbl

[18] M. Nagata, A general theory of algebraic geometry over Dedekind domains I, Am. J. Math., 78 (1956), 78-116. | MR | Zbl

[19] M. Nagata, Local rings, Interscience Tracts in Pure. Appl. Math, no. 13, Interscience, New York, (1962). MR 27 ll--5790. | MR | Zbl

[20] A. R. Wadsworth, Pairs of domains where all intermediate domains are Noetherian, Tran. Amer. Math. Soc., 195 (1974), 201-211. | MR | Zbl