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On the Ring of Continuous Functions with Countable Values and Compact Support | ||
| Caspian Journal of Mathematical Sciences | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 28 مهر 1404 | ||
| نوع مقاله: Research Articles | ||
| شناسه دیجیتال (DOI): 10.22080/cjms.2025.29848.1769 | ||
| نویسندگان | ||
| Mehrdad namdari* 1؛ Somayeh Soltanpour2؛ Zahra Alamneshan3 | ||
| 1Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran | ||
| 2Department of Science, Petroleum University of Technology, Ahvaz, Iran. | ||
| 3Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran. | ||
| تاریخ دریافت: 26 مرداد 1404، تاریخ بازنگری: 08 مهر 1404، تاریخ پذیرش: 21 مهر 1404 | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Projective؛ pure؛ pp-ring؛ $c$-basically disconnected | ||
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