Annihilator of ideals in $C(X)$

Mohamadian, Rostam

doi:10.22055/jamm.2025.47618.2300

Annihilator of ideals in $C(X)$

Document Type : Original Paper

Author

Rostam Mohamadian

Shahid Chamran University of Ahvaz

10.22055/jamm.2025.47618.2300

Abstract

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.

Keywords