Relative \(z\)-ideals in \(C(X)\). (English) Zbl 1167.54005

Azarpanah, F.; Taherifar, A.

Topology Appl. 156, No. 9, 1711-1717 (2009).

Let \(X\) be a completely regular Hausdorff space and \(C(X)\) denote the ring of all continuous real valued functions on \(X\). For any two ideals \(I\subset J\) in \(C(X)\) we say that \(I\) is a \(z_J\)-ideal in case, for all \(f \in I\) and \(g\in J\), \(Z(f)\subset Z(g)\) implies \(g\in I\). We say that \(I\) is a relative \(z\)-ideal, briefly a rez-ideal, in case there exists \(J\supset I\) such that \(J\neq i\) and \(I\) is a \(z_J\)-ideal. The present authors extend the work of F. Azarpanah and R. Mohamadian [Acta Math. Sin., Engl. Ser. 23, No. 6, 989–996 (2007; Zbl 1186.54021)] on the study of \(z_J\)- and res-ideals in \(C(X)\). They show, among other things, that, for any ideal \(I\) in \(C(X)\), every sub-ideal of \(I\) is a \(z_I\)-ideal if, and only if, \(Z(f)\) is open for all \(f\in I\). We recall that \(X\) is an \(F\)-space in case all cozero sets are \(C^*\)-embedded in \(X\). Among their other results are (i) \(X\) is an \(F\)-space if, and only if, for every ideal \(J\) in \(C(X)\), the sum of every two \(z_J\)-ideals is a \(z_J\)-ideal; (ii) \(X\) is a \(P\)-space if, and only if, every ideal in \(C(X)\) is a rez-ideal.

Reviewer: James V. Whittaker (Vancouver)

MSC:

54C40

Algebraic properties of function spaces in general topology

Keywords:

relative \(z\)-ideal; \(F\)-space; almost \(P\)-space; almost \(z\)-ideal

Citations:

Zbl 1186.54021

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