Some properties of the ideal of continuous functions with pseudocompact support
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-idea...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201010389 |
| Summary: | 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)={f∈C(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 coz I 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. |
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| ISSN: | 0161-1712 1687-0425 |