A class of ideals in intermediate rings of continuous functions

Authors

  • Sagarmoy Bag University of Calcutta
  • Sudip Kumar Acharyya University of Calcutta
  • Dhananjoy Mandal University of Calcutta

DOI:

https://doi.org/10.4995/agt.2019.10171

Keywords:

P-space, almost P-space, UMP-space, z-ideal, zâ—¦-ideal, ƷA-ideal

Abstract

For  any  completely  regular  Hausdorff  topological  space X,  an  intermediate  ring A(X) of  continuous  functions  stands  for  any  ring  lying between C∗(X) and C(X).  It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of zâ—¦-ideals.  It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X.  Interrelation between z-ideals, zâ—¦-ideal and ƷA-ideals in A(X) are examined.  It is proved that within the family of almost P-spaces X, each ƷA -ideal in A(X) is a zâ—¦-ideal if and only if each z-ideal in A(X) is a zâ—¦-ideal if and only if A(X) = C(X).

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Author Biographies

Sagarmoy Bag, University of Calcutta

Department of Pure Mathematics

Sudip Kumar Acharyya, University of Calcutta

Department of Pure Mathematics

Dhananjoy Mandal, University of Calcutta

Department of Pure Mathematics

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Published

2019-04-01

How to Cite

[1]
S. Bag, S. K. Acharyya, and D. Mandal, “A class of ideals in intermediate rings of continuous functions”, Appl. Gen. Topol., vol. 20, no. 1, pp. 109–117, Apr. 2019.

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