Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Connected Components of the Projective Line over a Ring

Andrea Blunck, Hans Havlicek

Abstract

The main result of the present paper is that the projective line over a ring $R$ is connected with respect to the relation "distant" if, and only if, $R$ is a $GE_2$-ring.

The Connected Components of the Projective Line over a Ring

Abstract

The main result of the present paper is that the projective line over a ring is connected with respect to the relation "distant" if, and only if, is a -ring.

Paper Structure

This paper contains 5 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Connected Components
  4. Generalized Chain Geometries
  5. Examples

Key Result

Lemma 3.1

Let $X'\in{\mathrm{GL}}_2(R)$ and suppose that the $2\times 2$-matrix $X$ over $R$ has the same first row as $X'$. Then $X$ is invertible if, and only if, there is a matrix such that $X = MX'$.

Theorems & Definitions (8)

  • Lemma 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 5.1
  • Theorem 5.3
  • Remark 5.4
  • Theorem 5.5
  • Remark 5.6