On distant-isomorphisms of projective lines

TL;DR

This paper classifies all distant-isomorphisms between projective lines over semilocal rings, revealing their connection to Jordan ring isomorphisms and projectivities, especially for certain semisimple rings.

Contribution

It provides a complete characterization of distant-isomorphisms over semilocal rings, linking them to Jordan isomorphisms and projectivities, and introduces a correspondence with Segre products of Grassmann spaces.

Findings

All distant-isomorphisms over certain semisimple rings are derived from Jordan isomorphisms and projectivities.

A one-to-one correspondence exists between projective lines over semisimple rings and Segre products of Grassmann spaces.

The classification applies particularly to semisimple rings without simple components isomorphic to fields.

Abstract

We determine all distant-isomorphisms between projective lines over semilocal rings. In particular, for those semisimple rings that do not have a simple component which is isomorphic to a field, every distant isomorphism arises from a Jordan isomorphism of rings and a projectivity. We show this by virtue of a one-one correspondence linking the projective line over a semisimple ring with a Segre product of Grassmann spaces.