International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 3, Pages 169-176

Some properties of the ideal of continuous functions with pseudocompact support

E. A. Abu Osba1 and H. Al-Ezeh2

1Department of Mathematics, University of Petra, Amman 961343, Jordan
2Department of Mathematics, University of Jordan, Amman 11942, Jordan

Received 23 April 2000; Revised 30 August 2000

Copyright © 2001 E. A. Abu Osba and H. Al-Ezeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let C(X) be the ring of all continuous real-valued functions defined on a completely regular T1-space. Let CΨ(X) and CK(X) be the ideal of functions with pseudocompact support and compact support, respectively. Further equivalent conditions are given to characterize when an ideal of C(X) is a P-ideal, a concept which was originally defined and characterized by Rudd (1975). We used this new characterization to characterize when CΨ(X) is a P-ideal, in particular we proved that CK(X) is a P-ideal if and only if CK(X)={fC(X):f=0 except on a finite set}. We also used this characterization to prove that for any ideal I contained in CΨ(X), I is an injective C(X)-module if and only if cozI is finite. Finally, we showed that CΨ(X) cannot be a proper prime ideal while CK(X) is prime if and only if X is an almost compact noncompact space and is an F-point. We give concrete examples exemplifying the concepts studied.