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

TL;DR

This paper explores the structure of rings of measurable functions, characterizing their maximal ideals, ultrafilters, and topological properties, especially focusing on Gelfand and Von-Neumann regular rings.

Contribution

It establishes conditions under which the space of ultrafilters and maximal ideals are homeomorphic, and characterizes regularity and Gelfand properties in these rings.

Findings

Homeomorphism between ultrafilter space and maximal ideals for Gelfand rings

Characterization of Von-Neumann regular rings via $ ext{Z}_S$-ideals

Introduction of $u_ ext{μ}$-topology and $m_ ext{μ}$-topology on the ring

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.