\(z\)-ideals and \(z^0\)-ideals in the factor rings of \(C(X)\)
An ideal \(I\) in \(C(X)\) is a \(z\)-ideal (\(z^0\)-ideal) if \(f \in I\), \(g \in C(X)\), and \(Z(f)\subseteq Z(g)\) (\(int_XZ(f) \subseteq int_XZ(X)\)) implies that \(g \in I\), \(Z(h)\) being the set of zeros of \(h\). As a result, \(I\) is a \(z\)-ideal (\(z^0\)-ideal) if and only if every prime ideal minimal over \(I\) is a \(z\)-ideal (\(z^0\)-ideal) [see \textit{F. Azarpanah} and \textit{R. Mohamadian}, Acta Math. Sin., Engl. Ser. 23, No. 6, 989--996 (2007; Zbl 1186.54021)]. The authors characterize \(z\)-ideals of the factor rings of \(C(X)\) via \(z\)-ideals of \(C(X)\). A family \(\{N_\pi(f) : f \in C(X), \pi \in C(X) \text{ is a positive unit}\}\) used as a base of open sets gives the \(m\)-topology on \(C(X)\). When \(X\) is pseudocompact, the paper shows that \(J/I\) is a \(z\)-ideal in \(C(X)/I\) if and only if \(J\) is a \(z\)-ideal in \(C(X)\) containing the \(m\)-closure \(\bar{I}\) of the ideal \(I\). The \(e\)-ideals in \(C^*(X)\) are the closed ideals with uniform norm topology, and the relative \(m\)-topology on \(C^*(X)\) coincides with the uniform norm topology when \(X\) is pseudocompact [see \textit{L. Gillman} and \textit{M. Jerison}, Rings of continuous functions. Reprint of the 1960 Van Nostrand edition. New York - Heidelberg - Berlin: Springer-Verlag (1976; Zbl 0327.46040)]. The paper shows that the sum of two \(m\)-closed ideals (\(e\)-ideals) in \(C(X)\) when \(X\) is pseudocompact is an \(m\)-closed ideal (\(e\)-ideal). Further, if \(I\) and \(J\) are \(z^0\)-ideals in \(C(X)\) with \(I \subseteq J\), then \(J/I\) is a \(z^0\)-ideal in \(C(X)/I\) if and only if every prime \(z^0\)-ideal in \(C(X)\) is minimal. Examples and counterexamples are given to illustrate some of the statements in their investigations of \(z^0\)-ideals of factor rings of \(C(X)\).
scientific article; zbMATH DE number 5901795