Irreducibility of ideals in a one-dimensional analytically irreducible ring

Valentina Barucci; Faten Khouja

Displaying similar documents to “Irreducibility of ideals in a one-dimensional analytically irreducible ring”

On $a$ -Kasch spaces

Ali Akbar Estaji, Melvin Henriksen (2010)

Archivum Mathematicum

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If $X$ is a Tychonoff space, $C (X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C (X)$ . Let $a$ be a infinite cardinal, then $X$ is called $a$ -Kasch (resp. $\bar{a}$ -Kasch) space if given any ideal (resp. $z$ -ideal) $I$ with $gen (I) < a$ then $I$ is a non-essential ideal. We show that $X$ is an $ℵ_{0}$ -Kasch space if and only if $X$ is an almost $P$ -space and $X$ is an $ℵ_{1}$ -Kasch space if and only if $X$ is a pseudocompact and almost $P$ -space. Let $C_{F} (X)$ denote the socle of $C (X)$ . For a topological space...

Intersections of essential minimal prime ideals

A. Taherifar (2014)

Commentationes Mathematicae Universitatis Carolinae

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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 $I \subseteq 𝒵 (ℛ)$ and $I$ is contained in no minimal prime ideal. We denote by $R_{K} (ℳ)$ , the set of all $a \in R$ for which $\bar{D (a)} = \bar{ℳ ∖ V (a)}$ 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...

Fixed-place ideals in commutative rings

Ali Rezaei Aliabad, Mehdi Badie (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let $I$ be a semi-prime ideal. Then $P_{\circ} \in Min (I)$ is called irredundant with respect to $I$ if $I \neq ⋂_{P_{\circ} \neq P \in Min (I)} P$ . 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 $p \in β X$ is a fixed-place point if $O^{p} (X)$ is a fixed-place ideal. In...

On nonregular ideals and $z^{\circ}$ -ideals in $C (X)$

F. Azarpanah, M. Karavan (2005)

Czechoslovak Mathematical Journal

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The spaces $X$ in which every prime $z^{\circ}$ -ideal of $C (X)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$ , such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^{\circ}$ -ideal in $C (X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C (X)$ a $z^{\circ}$ -ideal? When is every nonregular (prime) $z$ -ideal in $C (X)$ a...

On spaces with the ideal convergence property

Jakub Jasinski, Ireneusz Recław (2008)

Colloquium Mathematicae

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Let I ⊆ P(ω) be an ideal. We continue our investigation of the class of spaces with the I-ideal convergence property, denoted (I). We show that if I is an analytic, non-countably generated P-ideal then (I) ⊆ s₀. If in addition I is non-pathological and not isomorphic to $I_{b}$ , then (I) spaces have measure zero. We also present a characterization of the (I) spaces using clopen covers.

Displaying similar documents to “Irreducibility of ideals in a one-dimensional analytically irreducible ring”

On a -Kasch spaces

Intersections of essential minimal prime ideals

Fixed-place ideals in commutative rings

On nonregular ideals and z ∘ -ideals in C ( X )

On spaces with the ideal convergence property

On $a$ -Kasch spaces

On nonregular ideals and $z^{\circ}$ -ideals in $C (X)$