A generalized version of the rings CK(X) and C∞(X)– an enquery about when they become Noetheri
Submitted: 2014-08-29
|Accepted: 2015-01-09
|Published: 2015-02-10
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Keywords:
Noetherian ring, Artinian ring, totally ordered field, zero-dimensional space, pseudocompact support, relatively pseudocompact support
Supporting agencies:
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.
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