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