A generalized version of the rings CK (X) and C∞(X)– an enquery about when they become Noetheri

[EN] Suppose F is a totally ordered field equipped with its order topology and X a completely F-regular topological space. Suppose P is an ideal of closed sets in X and X is locally-P. Let CP(X, F) = {f : X ! F | f is continuous on X and its support belongs to P} and CP∞ (X, F) = {f 2 CP(X, F) | 8" > 0 in F, clX{x 2 X : |f(x)| > "} 2 P}. Then CP(X, F) is a Noetherian ring if and only if CP∞ (X, F) is a Noetherian ring if and only if X is a finite set. The fact that a locally compact Hausdorff space X is finite if and only if the ring CK(X) is Noetherian if and only if the ring C∞(X) is Noetherian, follows as a particular case on choosing F = R and P = the ideal of all compact sets in X. On the other hand if one takes F = R and P = the ideal of closed relatively pseudocompact subsets of X, then it follows that a locally pseudocompact space X is finite if and only if the ring C (X) of all real valued continuous functions on X with pseudocompact support is Noetherian if and only if the ring C ∞ (X) = {f 2 C(X) | 8" > 0, clX{x 2 X : |f(x)| > "} is pseudocompact } is Noetherian. Finally on choosing F = R and P = the ideal of all closed sets in X, it follows that: X is finite if and only if the ring C(X) is Noetherian if and only if the ring C∗(X) is Noetherian.The third author thanks the UGC, New Delhi-110002, India, for financial supportAcharyya, SK.; Chattopadhyay, KC.; Rooj, P. (2015). A generalized version of the rings CK (X) and C∞(X)– an enquery about when they become Noetheri. Applied General Topology. 16(1):81-87. doi:10.4995/agt.2015.32478187161S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of subsets of X, Topology Proc. 35 (2010), 127-148.I. Gelfand and A. Kolmogoroff, On Rings of Continuous Functions on topological spaces, Dokl. Akad. Nauk SSSR 22 (1939), 11-15.C. W. Kohls, Ideals in rings of Continuous Functions, Fund. Math. 45 (1957), 28-50.C. W. Kohls, Prime ideals in rings of Continuous Functions, Illinois. J. Math. 2 (1958), 505-536