{"title":"Structure spaces and allied problems on a class of rings of measurable functions","authors":"Soumajit Dey, Sudip Kumar Acharyya, Dhananjoy Mandal","doi":"arxiv-2408.00505","DOIUrl":null,"url":null,"abstract":"A ring $S(X,\\mathcal{A})$ of real valued $\\mathcal{A}$-measurable functions\ndefined over a measurable space $(X,\\mathcal{A})$ is called a $\\chi$-ring if\nfor each $E\\in \\mathcal{A} $, the characteristic function $\\chi_{E}\\in\nS(X,\\mathcal{A})$. The set $\\mathcal{U}_X$ of all $\\mathcal{A}$-ultrafilters on\n$X$ with the Stone topology $\\tau$ is seen to be homeomorphic to an appropriate\nquotient space of the set $\\mathcal{M}_X$ of all maximal ideals in\n$S(X,\\mathcal{A})$ equipped with the hull-kernel topology $\\tau_S$. It is\nrealized that $(\\mathcal{U}_X,\\tau)$ is homeomorphic to\n$(\\mathcal{M}_S,\\tau_S)$ if and only if $S(X,\\mathcal{A})$ is a Gelfand ring.\nIt is further observed that $S(X,\\mathcal{A})$ is a Von-Neumann regular ring if\nand only if each ideal in this ring is a $\\mathcal{Z}_S$-ideal and\n$S(X,\\mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a\n$\\mathcal{Z}_S$-ideal. A pair of topologies $u_\\mu$-topology and\n$m_\\mu$-topology, are introduced on the set $S(X,\\mathcal{A})$ and a few\nproperties are studied.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.00505","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.