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

Abstract

[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.