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

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 C_P(X,F) ={f:X→F|f is continuous on X and its support belongs to P} and C^P_∞(X,F) ={f∈C_P(X,F)| ∀ε>0 in F, cl_X{x∈X:|f(x)|> ε} ∈ P}. Then C_P(X,F) is a Noetherian ring if and only if C^P_∞ (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 C_K(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_ψ(X) of  all  real  valued  continuous functions on X with pseudocompact support is Noetherian if and only if  the ring C^ψ_∞(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.