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

Résumé

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.