On the ring of continuous functions with countable values and compact support

Namdari, Mehrdad; Soltanpour, Somayeh; Alamneshan, Zahra

doi:10.22080/cjms.2025.29848.1769

On the ring of continuous functions with countable values and compact support

Document Type : Research Articles

Authors

¹ Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

² Department of Science, Petroleum University of Technology, Ahvaz, Iran.

³ Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

10.22080/cjms.2025.29848.1769

Abstract

In this paper, we investigate the structure of C_{cK}(X), the set of all functions f \in C_c(X) whose support, defined as \operatorname{cl}_X(X \setminus Z(f)), is compact. We study C_{cK}(X) as an ideal of C_c(X)$ and characterize its closure in the topological ring C_{c_m}(X) as the intersection of all maximal ideals containing it. Additionally, we introduce the space X_{cL} and examine its relationship with C_{cK}(X), particularly in connection with the purity and projectivity of the ideal. We establish necessary and sufficient conditions for C_{cK}(X) to be a pure or projective C_c(X)-module. Moreover, we show that C_c(X) is a pp-ring if and only if the space X is c-basically disconnected. Finally, we prove that C_{cK}(X) is a pure ideal and that X_{cL} is c-basically disconnected if and only if every principal ideal (f), with f \in C_{cK}(X), is a projective C_c(X)-module.

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