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 -ring if for each E A, the characteristic function ₄ S (X, A). The set UX of all A-ultrafilters on X with the Stone topology 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 S. It is realized that (UX, ) is homeomorphic to (MS, S) 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_-topology and m_-topology, are introduced on the set S (X, A) and a few properties are studied.