\(z\)-ideals and \(z^0\)-ideals in the factor rings of \(C(X)\). (English) Zbl 1221.54023

Aliabad, A. R.; Azarpanah, F.; Paimann, M.

Bull. Iran. Math. Soc. 36, No. 1, 211-226 (2010).

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 F. Azarpanah and 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 L. Gillman and 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 \(C(X)\).

Reviewer: Pao-Sheng Hsu (Columbia Falls)

MSC:

54C40	Algebraic properties of function spaces in general topology
13A15	Ideals and multiplicative ideal theory in commutative rings

Keywords:

reduced ring; \(z\)-ideal; \(z^0\)-ideals; \(m\)-topology; closed ideals; \(P\)-space; pseudocompact; \(\varepsilon\)-set

Citations:

Zbl 1186.54021; Zbl 0327.46040