Annihilator of ideals in $C(X)$

Mohamadian, Rostam

doi:10.22055/jamm.2025.47618.2300

Annihilator of ideals in $C(X)$

نوع مقاله : مقاله پژوهشی

نویسنده

Rostam Mohamadian

دانشگاه شهید چمران اهواز

10.22055/jamm.2025.47618.2300

چکیده

Let $I$ be an ideal of $C(X)$. In this paper we show that: a) ${\rm Ann}(I)=O^{\beta X\setminus \theta(I)}$, b) ${\rm Ann(Ann}(I))=O^{int_{\beta X}\theta(I)}$, c) $m{\rm Ann(Ann}(I))=O^{cl_{\beta X}int_{\beta X}\theta(I)}$, where $\theta(I)=\bigcap_{f\in I}cl_{\beta X}Z(f)$ and $mI$ be the pure part of $I$. It is also show that $X$ is a basically disconnected space if and only if for every $f\in C(X)$ and every ideal $I$ of $C(X)$, ${\rm Ann}(f)+{\rm Ann}(I)= {\rm Ann}(fI)$.

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توپولوژی

عنوان مقاله [English]

Annihilator of ideals in $C(X)$

نویسنده [English]

Rostam Mohamadian

Shahid Chamran University of Ahvaz

چکیده [English]

Let $I$ be an ideal of $C(X)$. In this paper we show that ${\rm Ann}(I)=O^{\beta X\setminus \theta(I)}$ and $m{\rm Ann}(I)=O^{{\beta X}\setminus int_{\beta X}\theta(I)}$, where $\theta(I)=\bigcap_{f\in I}cl_{\beta X}Z(f)$ and $mI$ is the pure part of $I$. We also show that
${\rm Ann(Ann}(I))=O^{int_{\beta X}\theta(I)}$ and $m{\rm Ann(Ann}(I))=O^{cl_{\beta X}int_{\beta X}\theta(I)}$. Finally we show that a space $X$ is a $\partial$-space if and only if every nonregular prime ideal of $C(X)$ is a $z$-ideal.

کلیدواژه‌ها [English]