Fixed-place ideals in commutative rings

Ali Rezaei Aliabad; Mehdi Badie

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 1, page 53-68
  • ISSN: 0010-2628

Abstract

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Let I be a semi-prime ideal. Then P ∘ ∈ Min ( I ) is called irredundant with respect to I if I ≠ ⋂ P ∘ ≠ P ∈ 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 ∈ β 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- 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- rank X ( p ) = η if and only if there is a family { Y α } α ∈ A of cozero sets of X such that: 1- | A | = η , 2- p ∈ cl β X Y α for each α ∈ A , 3- p ∉ cl β X ( Y α ∩ Y β ) if α ≠ β and 4- η is the greatest cardinal with the above properties. In this case p is an F -point with respect to Y α for any α ∈ A .

How to cite

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Aliabad, Ali Rezaei, and Badie, Mehdi. "Fixed-place ideals in commutative rings." Commentationes Mathematicae Universitatis Carolinae 54.1 (2013): 53-68. <http://eudml.org/doc/252487>.

@article{Aliabad2013,
abstract = {Let $I$ be a semi-prime ideal. Then $P_\circ \in \operatorname\{Min\}(I)$ is called irredundant with respect to $I$ if $I\ne \bigcap _\{P_\circ \ne P\in \operatorname\{Min\}(I)\}P$. 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 $\lbrace Y_\alpha \rbrace _\{ \alpha \in A\}$ 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 \ne \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 $\alpha \in A$.},
author = {Aliabad, Ali Rezaei, Badie, Mehdi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ring of continuous functions; fixed-place; anti fixed-place; irredundant ideal; semi-prime; annihilator; affiliated prime; fixed-place rank; Zariski topology; fixed-place; anti fixed-place; irredundant ideal; annihilator; affiliated prime; Zariski topology},
language = {eng},
number = {1},
pages = {53-68},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Fixed-place ideals in commutative rings},
url = {http://eudml.org/doc/252487},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Aliabad, Ali Rezaei
AU - Badie, Mehdi
TI - Fixed-place ideals in commutative rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 1
SP - 53
EP - 68
AB - Let $I$ be a semi-prime ideal. Then $P_\circ \in \operatorname{Min}(I)$ is called irredundant with respect to $I$ if $I\ne \bigcap _{P_\circ \ne P\in \operatorname{Min}(I)}P$. 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 $\lbrace Y_\alpha \rbrace _{ \alpha \in A}$ 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 \ne \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 $\alpha \in A$.
LA - eng
KW - ring of continuous functions; fixed-place; anti fixed-place; irredundant ideal; semi-prime; annihilator; affiliated prime; fixed-place rank; Zariski topology; fixed-place; anti fixed-place; irredundant ideal; annihilator; affiliated prime; Zariski topology
UR - http://eudml.org/doc/252487
ER -

References

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