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 $\chi$-ring if for each $E\in \mathcal{A}$, the characteristic function ${\chi }_E\in S(X,\mathcal{A})$. The set ${\mathcal{U}}_X$ of all $\mathcal{A}$-ultrafilters on $X$ with the Stone topology $\tau$ 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 ${\tau }_S$. It is realized that $({\mathcal{U}}_X,\tau )$ is homeomorphic to $({\mathcal{M}}_S,{\tau }_S)$ 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 $u_{\mu }$-topology and $m_{\mu }$-topology, are introduced on the set $S(X,\mathcal{A})$ and a few properties are studied.