{
  "id": "1803.06271",
  "version": "v1",
  "published": "2018-03-16T15:33:14.000Z",
  "updated": "2018-03-16T15:33:14.000Z",
  "title": "Maximal ideals in rings of real measurable functions",
  "authors": [
    "Ali Akbar Estaji",
    "Ahmad Mahmoudi Darghadam",
    "Hasan Yousefpour"
  ],
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "Let $ M (X)$ be the ring of all real measurable functions on a measurable space $(X, \\mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\\circ}$-ideal. Also, we give several characterizations of maximal ideals of $M(X)$, mostly in terms of certain lattice-theoretic properties of $\\mathscr{A}$. The notion of $T$-measurable space is introduced. Next, we show that for every measurable space $(X,\\mathscr{A})$ there exists a $T$-measurable space $(Y,\\mathscr{A}^{\\prime})$ such that $M(X)\\cong M(Y)$ as rings. The notion of compact measurable space is introduced. Next, we prove that if $(X, \\mathscr{A})$ and $(Y, \\mathfrak{M^{\\prime}})$ are two compact $T$-measurable spaces, then $X\\cong Y$ as measurable spaces if and only if $M(X)\\cong M (Y)$ as rings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-16T15:33:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A20",
      "13A30",
      "54C30",
      "06D22"
    ],
    "keywords": [
      "real measurable functions",
      "maximal ideals",
      "lattice-theoretic properties",
      "compact measurable space",
      "characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}