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

Abstract

A ring $S(X,\mathcal{A})$ of real valued $\mathcal{A}$-measurable functions defined over a measurable space $(X,\mathcal{A})$ is called a $χ$-ring if for each $E\in \mathcal{A} $, the characteristic function $χ_{E}\in S(X,\mathcal{A})$. The set $\mathcal{U}_X$ of all $\mathcal{A}$-ultrafilters on $X$ with the Stone topology $τ$ is seen to be homeomorphic to an appropriate quotient space of the set $\mathcal{M}_X$ of all maximal ideals in $S(X,\mathcal{A})$ equipped with the hull-kernel topology $τ_S$. It is realized that $(\mathcal{U}_X,τ)$ is homeomorphic to $(\mathcal{M}_S,τ_S)$ if and only if $S(X,\mathcal{A})$ is a Gelfand ring. It is further observed that $S(X,\mathcal{A})$ is a Von-Neumann regular ring if and only if each ideal in this ring is a $\mathcal{Z}_S$-ideal and $S(X,\mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a $\mathcal{Z}_S$-ideal. A pair of topologies $u_μ$-topology and $m_μ$-topology, are introduced on the set $S(X,\mathcal{A})$ and a few properties are studied.