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On the direct product of fields with an application

Abolfazl Tarizadeh

Abstract

In this paper, the (infinite) direct product of fields is investigated. In particular, the finiteness of a given set is characterized in terms of some ring-theoretic observations. Next, a certain localization (whose multiplicative set formed by cofinite sets) of the direct product of fields is studied. Finally, it is shown that every set $X$ can be made into a separated scheme, and this scheme is an affine scheme if and only if $X$ is a finite set.

On the direct product of fields with an application

Abstract

In this paper, the (infinite) direct product of fields is investigated. In particular, the finiteness of a given set is characterized in terms of some ring-theoretic observations. Next, a certain localization (whose multiplicative set formed by cofinite sets) of the direct product of fields is studied. Finally, it is shown that every set can be made into a separated scheme, and this scheme is an affine scheme if and only if is a finite set.

Paper Structure

This paper contains 4 sections, 10 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some observations on the ring $\Lambda$
  4. A geometric result

Key Result

Theorem 3.1

If $f,g\in\Lambda$ then the following assertions hold. $\mathbf{(i)}$$(f)=\{h\in\Lambda: \operatorname{S}(h)\subseteq\operatorname{S}(f)\}$ and the ring $\Lambda/(f)$ is canonically isomorphic to $\prod\limits_{x\in X\setminus\operatorname{S}(f)}K_{x}$. $\mathbf{(ii)}$$(f)=(g)$ if and only if $\oper

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • proof
  • ...and 10 more