Intermediate domains between a domain and some intersection of its localizations

Ben Nasr, Mabrouk ; Jarboui, Noômen

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 701-713 / Harvested from Biblioteca Digitale Italiana di Matematica

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.

Résumé

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.

Détail

Publié le : 2002-10-01

MR 1934375 | Zbl 1177.13016

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