The Connected Components of the Projective Line over a Ring

TL;DR

This paper characterizes the connectedness of the projective line over a ring in terms of the ring's algebraic property of being a $GE_2$-ring, linking geometric and algebraic structures.

Contribution

It establishes a precise criterion for the connectedness of the projective line over a ring based on the ring's $GE_2$-property, a new connection between geometry and algebra.

Findings

The projective line over a ring is connected iff the ring is a $GE_2$-ring.

Provides a characterization linking geometric connectedness to algebraic properties.

Enhances understanding of the structure of projective lines over rings.

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.