A class of ideals in intermediate rings of continuous functions

Sagarmoy Bag, Sudip Kumar Acharyya, Dhananjoy Mandal

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 z◦-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, z◦-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 z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).


Keywords

P-space; almost P-space; UMP-space; z-ideal; z◦-ideal; ƷA-ideal

Subject classification

Primary 54C40; Secondary 46E25

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Universitat Politècnica de València

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