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Minimal and maximal spectra as the Stone-Čech compactification

A. Tarizadeh, M. R. Rezaee

Abstract

In this paper, new advances on the compactifications of topological spaces, especially on the Stone-Čech and Alexandroff compactifications have been made. Among the main results, it is proved that the minimal spectrum of the direct product of a family of integral domains indexed by a set $X$ is the Stone-Čech compactification of the discrete space $X$. Dually, it is proved that the maximal spectrum of the direct product of a family of local rings indexed by $X$ is also the Stone-Čech compactification of the discrete space $X$. The Alexandroff (one-point) compactification of a discrete space is constructed by a new method. Next, we proceed to give a natural and quite simple way to construct ultra-rings. Then this new approach is used to obtain several new results on the Stone-Čech compactification.

Minimal and maximal spectra as the Stone-Čech compactification

Abstract

In this paper, new advances on the compactifications of topological spaces, especially on the Stone-Čech and Alexandroff compactifications have been made. Among the main results, it is proved that the minimal spectrum of the direct product of a family of integral domains indexed by a set is the Stone-Čech compactification of the discrete space . Dually, it is proved that the maximal spectrum of the direct product of a family of local rings indexed by is also the Stone-Čech compactification of the discrete space . The Alexandroff (one-point) compactification of a discrete space is constructed by a new method. Next, we proceed to give a natural and quite simple way to construct ultra-rings. Then this new approach is used to obtain several new results on the Stone-Čech compactification.

Paper Structure

This paper contains 10 sections, 40 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal spectrum as Stone-Čech compactification
  4. The Stone-Čech compactification of an arbitrary space
  5. Maximal spectrum as Stone-Čech compactification
  6. Ultra-rings and their applications in compactification
  7. Alexandroff compactification
  8. Totally disconnected compactifications
  9. Semigroup structure on $\beta X$
  10. Absolutely flatness of the total ring of fractions

Key Result

Proposition 3.1

Consider the canonical ring map $\pi:R\rightarrow S^{-1}R$ where $S$ is a multiplicative subset of a ring $R$, and let $f\in R$. Then $f\in\bigcap\limits_{\mathfrak{p}\in\operatorname{Im}\pi^{\ast}}\mathfrak{p}$ if and only if there exists some $g\in S$ such that $fg$ is nilpotent.

Theorems & Definitions (87)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • ...and 77 more