Fixed-place ideals in commutative rings

Aliabad, Ali Rezaei ; Badie, Mehdi

Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 1, pp. 53-68.

Résumé

Let $I$ be a semi-prime ideal. Then $P_\circ \in \operatorname{Min}(I)$ is called irredundant with respect to $I$ if $I\neq \bigcap_{P_\circ \neq P\in \operatorname{Min}(I)}P$. If $I$ is the intersection of all irredundant ideals with respect to $I$, it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$, it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p\in \beta X$ is a fixed-place point if $O^p(X)$ is a fixed-place ideal. In this situation the fixed-place rank of $p$, denoted by FP-$\operatorname{rank}_X(p)$, is defined as the cardinal of the set of all irredundant prime ideals with respect to $O^p(X)$. Let $p$ be a fixed-place point, it is shown that FP-$\operatorname{rank}_X (p)= \eta $ if and only if there is a family $\{Y_\alpha\}_{ \alpha \in A}$ of cozero sets of $X$ such that: 1- $|A|= \eta $, 2- $p\in \operatorname{cl}_{\beta X} Y_\alpha$ for each $\alpha \in A$, 3- $p\notin \operatorname{cl}_{\beta X} (Y_\alpha \cap Y_\beta )$ if $\alpha \neq \beta $ and 4- $\eta $ is the greatest cardinal with the above properties. In this case $p$ is an $F$-point with respect to $Y_\alpha $ for any $\alpha \in A$.

Détail

Classification : 13Axx, 54C40
Keywords: ring of continuous functions; fixed-place; anti fixed-place; irredundant; semi-prime; annihilator; affiliated prime; fixed-place rank; Zariski topology