Structure spaces and allied problems on a class of rings of measurable functions

A ring S(X,A) of real valued A-measurable functions defined over a measurable space (X,A) is called a χ-ring if for each E∈ A, the characteristic function χE∈ S(X,A). The set UX of all A-ultrafilters on X with the Stone topology τ is seen to be homeomorphic to an appropriate quotient space of the set MX of all maximal ideals in S(X,A) equipped with the hull-kernel topology τS. It is realized that (UX,τ) is homeomorphic to (MS,τS) if and only if S(X,A) is a Gelfand ring. It is further observed that S(X,A) is a Von-Neumann regular ring if and only if each ideal in this ring is a ZS-ideal and S(X,A) is Gelfand when and only when every maximal ideal in it is a ZS-ideal. A pair of topologies u_μ-topology and m_μ-topology, are introduced on the set S(X,A) and a few properties are studied.