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

Authors

  • Sudip Kumar Acharyya University of Calcutta
  • Kshitish Chandra Chattopadhyay University of Burdwan
  • Pritam Rooj University of Calcutta

DOI:

https://doi.org/10.4995/agt.2015.3247

Keywords:

Noetherian ring, Artinian ring, totally ordered field, zero-dimensional space, pseudocompact support, relatively pseudocompact support

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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Author Biographies

Sudip Kumar Acharyya, University of Calcutta

Department of Pure Mathematics

Kshitish Chandra Chattopadhyay, University of Burdwan

Department of Mathematics

Pritam Rooj, University of Calcutta

Department of Pure Mathematics

References

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Published

2015-02-10

How to Cite

[1]
S. K. Acharyya, K. C. Chattopadhyay, and P. Rooj, “A generalized version of the rings CK(X) and C∞(X)– an enquery about when they become Noetheri”, Appl. Gen. Topol., vol. 16, no. 1, pp. 81–87, Feb. 2015.

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Regular Articles