 X
A generalized version of the rings CK(X) and Câˆž(x)â€“ an enquery about when they become noetheri
S K ACHARYYA, K C CHATTOPADHYAY, P ROOJ
Published in Universidad Politecnica de Valencia
2015
Volume: 16

Issue: 1
Pages: 81 - 87
Abstract
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 ∈ CP(X,F) |∀ε> 0 in F,clx{x ∈ X: |f(x)| > ε} ∈ 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 Cψ (X) of all real valued continuous functions on X with pseudocompact support is Noetherian if and only if the ring Cψ∞(X) = {f ∈ C(X)|∀ε > 0,clx{x ∈ X: |f(x)| > ε} 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. © AGT, UPV, 2015.