Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces

Echi, Othman

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 489-507

Résumé

In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if $g \colon Y \to X$ is a quasi-homeomorphism, $Z$ a sober space and $f \colon Y \to Z$ a continuous map, then there exists a unique continuous map $F \colon X \to Z$ such that $F \circ g =f$. Let $X$ be a $T_{0}$-space, $q \colon X \to^{s} X$ the injection of $X$ onto its sobrification $^{s}X$. It is shown, here, that $q(\text{Gold}(X))=\text{Gold}(\sideset{^{s}}{}{\operatorname{X}})$, where $\text{Gold}(X)$ is the set of all locally closed points of $X$. Some applications are also indicated. The Jacobson prime spectrum of a commutative ring $R$ is the set of all prime ideals of $R$ which are intersections of some maximal ideals of $R$. One of our main results is a surprising answer to the problem of ordered disjoint union of jacspectral sets (ordered sets which are isomorphic to the Jacobson prime spectrum of some ring): Let $\{(X_{\lambda}, \leq_{\lambda}) \, : \, \lambda\in\Lambda \}$ be a collection of ordered disjoint sets and $X=\bigcup_{\lambda\in\Lambda} X_{\lambda}$. Partially order $X$ by declaring $x\leq y$ to mean that there exists $\lambda\in\Lambda$ such that $x$, $y\in X_{\lambda}$ and $x\leq_{\lambda} y$. Then the following statements are equivalent: (i) $(X, \leq)$ is jacspectral. (ii) $(X_{\lambda}, \leq_{\lambda})$ is jacspectral, for each $\lambda\in\Lambda$.