Abstract
<p>Let SC<sub>F</sub>(X) denote the ideal of C(X) consisting of functions which are zero everywhere except on a countable number of points of X. It is generalization of the socle of C(X) denoted by C<sub>F</sub>(X). Using this concept we extend some of the basic results concerning C<sub>F</sub>(X) to SC<sub>F</sub>(X). In particular, we characterize the spaces X such that SC<sub>F</sub>(X) is a prime ideal in C(X) (note, C<sub>F</sub>(X) is never a prime ideal in C(X)). This may be considered as an advantage of SC<sub>F</sub>(X) over C(X). We are also interested in characterizing topological spaces X such that C<sub>c</sub>(X) =R+SC<sub>F</sub>(X), where C<sub>c</sub>(X) denotes the subring of C(X) consisting of functions with countable image.</p>
Publisher
Universitat Politecnica de Valencia
Reference17 articles.
1. F. Azarpanah, Algebraic properties of some compact spaces, Real Anal. Exchange 25 (2000), 317-328.
2. F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar. 31 (1995), 105-112. https://doi.org/10.1007/BF01876485
3. F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 2149-2154. https://doi.org/10.1090/S0002-9939-97-04086-0
4. F. Azarpanah and O. A. S. Karamzadeh, Algebric characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), 155-168.
5. F. Azarpanah, O. A. S. Karamzadeh and S. Rahmati, C(X) vs. C(X) modulo its socle, Coll. Math. 3 (2008), 315-336. https://doi.org/10.4064/cm111-2-9