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<p>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 Ʒ_A-ideals in A(X) are examined.  It is proved that within the family of almost P-spaces X, each Ʒ_A -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).</p>

The set of all maximal ideals of the ring M(X, A) of real valued measurable functions on a measurable space (X, A) equipped with the hullkernel topology is shown to be homeomorphic to the setX of all ultrafilters of measurable sets on X with the Stone-topology. This yields a complete description of the maximal ideals of M(X, A) in terms of the points ofX. It is further shown that the structure spaces of all the intermediate subrings of M(X, A) containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when X is a P -space, then C(X) = M(X, A) where A is the σ-algebra consisting of the zero-sets of X.2010 Mathematics Subject Classification. Primary 54C40; Secondary 46E30. Key words and phrases. Rings of Measurable functions, intermediate rings of measurable functions, A-filter on X, A-ultrafilter on X, A-ideal, absolutely convex ideals; hull-kernel topology; Stone-topology; conditionally complete lattice; P -space.The second author thanks the NBHM, Mumbai-400 001, India, for financial support.

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