Intersections of essential minimal prime ideals

A. Taherifar

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 1, page 121-130
  • ISSN: 0010-2628

Abstract

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Let 𝒵 ( ℛ ) be the set of zero divisor elements of a commutative ring R with identity and ℳ be the space of minimal prime ideals of R with Zariski topology. An ideal I of R is called strongly dense ideal or briefly s d -ideal if I ⊆ 𝒵 ( ℛ ) and I is contained in no minimal prime ideal. We denote by R K ( ℳ ) , the set of all a ∈ R for which D ( a ) ¯ = ℳ ∖ V ( a ) ¯ is compact. We show that R has property ( A ) and ℳ is compact if and only if R has no s d -ideal. It is proved that R K ( ℳ ) is an essential ideal (resp., s d -ideal) if and only if ℳ is an almost locally compact (resp., ℳ is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring R need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring R is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of C ( X ) is equal to the socle of C(X) (i.e., C F ( X ) = O β X ∖ I ( X ) ). Finally, we show that a topological space X is pseudo-discrete if and only if I ( X ) = X L and C K ( X ) is a pure ideal.

How to cite

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Taherifar, A.. "Intersections of essential minimal prime ideals." Commentationes Mathematicae Universitatis Carolinae 55.1 (2014): 121-130. <http://eudml.org/doc/260810>.

@article{Taherifar2014,
abstract = {Let $\mathcal \{Z(R)\}$ be the set of zero divisor elements of a commutative ring $R$ with identity and $\mathcal \{M\}$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq \mathcal \{Z(R)\}$ and $I$ is contained in no minimal prime ideal. We denote by $R_\{K\}(\mathcal \{M\})$, the set of all $a\in R$ for which $\overline\{D(a)\}= \overline\{\mathcal \{M\}\setminus V(a)\}$ is compact. We show that $R$ has property $(A)$ and $\mathcal \{M\}$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_\{K\}(\mathcal \{M\})$ is an essential ideal (resp., $sd$-ideal) if and only if $\mathcal \{M\}$ is an almost locally compact (resp., $\mathcal \{M\}$ is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring $R$ need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring $R$ is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of $C(X)$ is equal to the socle of C(X) (i.e., $C_\{F\}(X)= O^\{\beta X\setminus I(X)\}$). Finally, we show that a topological space $X$ is pseudo-discrete if and only if $I(X)=X_\{L\}$ and $C_\{K\}(X)$ is a pure ideal.},
author = {Taherifar, A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {essential ideals; $sd$-ideal; almost locally compact space; nowhere dense; Zariski topology; essential ideals; -ideal; almost locally compact space; nowhere dense; Zariski topology},
language = {eng},
number = {1},
pages = {121-130},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Intersections of essential minimal prime ideals},
url = {http://eudml.org/doc/260810},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Taherifar, A.
TI - Intersections of essential minimal prime ideals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 1
SP - 121
EP - 130
AB - Let $\mathcal {Z(R)}$ be the set of zero divisor elements of a commutative ring $R$ with identity and $\mathcal {M}$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq \mathcal {Z(R)}$ and $I$ is contained in no minimal prime ideal. We denote by $R_{K}(\mathcal {M})$, the set of all $a\in R$ for which $\overline{D(a)}= \overline{\mathcal {M}\setminus V(a)}$ is compact. We show that $R$ has property $(A)$ and $\mathcal {M}$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_{K}(\mathcal {M})$ is an essential ideal (resp., $sd$-ideal) if and only if $\mathcal {M}$ is an almost locally compact (resp., $\mathcal {M}$ is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring $R$ need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring $R$ is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of $C(X)$ is equal to the socle of C(X) (i.e., $C_{F}(X)= O^{\beta X\setminus I(X)}$). Finally, we show that a topological space $X$ is pseudo-discrete if and only if $I(X)=X_{L}$ and $C_{K}(X)$ is a pure ideal.
LA - eng
KW - essential ideals; $sd$-ideal; almost locally compact space; nowhere dense; Zariski topology; essential ideals; -ideal; almost locally compact space; nowhere dense; Zariski topology
UR - http://eudml.org/doc/260810
ER -

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