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

Sudip Kumar Acharyya

India

University of Calcutta

Department of Pure Mathematics

Kshitish Chandra Chattopadhyay

India

University of Burdwan

Department of Mathematics

Pritam Rooj

India

University of Calcutta

Department of Pure Mathematics
|

Accepted: 2015-01-09

|

Published: 2015-02-10

DOI: https://doi.org/10.4995/agt.2015.3247
Funding Data

Downloads

Keywords:

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

Supporting agencies:

This research was not funded

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.

Show more Show less

References:

S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of subsets of X, Topology Proc. 35 (2010), 127-148.

S. K. Acharyya, K. C. Chattopadhyay and P. P. Ghosh, Constructing the Banaschewski Compactification without the Dedekind completeness axiom, Int. J. Math. Math. Sciences 69 (2004), 3799-3816. https://doi.org/10.1155/S0161171204305168

D. S. Dummit and R. M. Foote, Abstract Algebra. 2nd Edition, John Wiley and Sons, Inc., 2005. https://doi.org/10.1002/div.3305

L. Gillman and M. Jerison, Rings of Continuous Functions. New York: Van Nostrand Reinhold Co., 1960. https://doi.org/10.1007/978-1-4615-7819-2

I. Gelfand and A. Kolmogoroff, On Rings of Continuous Functions on topological spaces, Dokl. Akad. Nauk SSSR 22 (1939), 11-15.

H. E. Hewitt, Rings of real valued Continuous Functions, I, Trans. Amer. Math. Soc. 64 (1948), 54-99. https://doi.org/10.1090/S0002-9947-1948-0026239-9

D. G. Johnson and M. Mandelkar, Functions with pseudocompact support, General Topology and its App. 3 (1973), 331-338. https://doi.org/10.1016/0016-660X(73)90020-2

C. W. Kohls, Ideals in rings of Continuous Functions, Fund. Math. 45 (1957), 28-50. https://doi.org/10.4064/fm-45-1-28-50

C. W. Kohls, Prime ideals in rings of Continuous Functions, Illinois. J. Math. 2 (1958), 505-536. https://doi.org/10.1215/ijm/1255454113

M. Mandelkar, Support of Continuous Functions, Trans. Amer. Math. Soc. 156 (1971), 73-83. https://doi.org/10.1090/S0002-9947-1971-0275367-4

M. H. Stone, Applications of the theory of Boolean rings to General Topology, Trans. Amer. Math. Soc. 41 (1937), 375-481. https://doi.org/10.1090/S0002-9947-1937-1501905-7

Show more Show less