$z^\circ$-filters and related ideals in $C(X)$

Document Type : Research Paper


Shahid Chamran University of Ahvaz


In this article we introduce the concept of $z^\circ$-filter on a topological space $X$. We study and investigate the behavior of $z^\circ$-filters and compare them  with corresponding ideals, namely, $z^\circ$-ideals of $C(X)$,  the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^\circ$-filter is ci-fixed. Finally, by using  $z^\circ$-ultrafilters, we prove that any arbitrary product of i-compact spaces is i-compact.


[1] F. Azarpanah, On almost P-spaces, Far East J. Math. Sci., Special Volume, 121{132 (2000).
[2] F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On z-ideals in C(X), Fund. Math., 160, 15{25 (1999).
[3] F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On ideal consisting entirely of zerodivisor, Comm. Algebra, 28(2), 1061{1073 (2000).
[4] F. Azarpanah and M. Karavan, On nonregular ideals and z-ideal in C(X), Cech. Math. J., 55(130), 397{407 (2005).
[5] F. Azarpanah and R. Mohamadian, pz-ideals and pz-ideals in C(X), Acta. Math. Sinica., English Series, 23,
989{996 (2007).
[6] R. Engelking, General Topology, PWN-Polish Sci Publ, 1977.
[7] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976.
[8] G. Mason, z-ideals and prime ideals, J. Algebra, 26: 280{297 (1973).