Free cyclic submodules in the context of the projective line

TL;DR

This paper studies free cyclic submodules over rings, focusing on those generated by outliers, and classifies their orbits in the ring of lower triangular 3x3 matrices over a field, revealing diverse behaviors.

Contribution

It provides a complete classification of orbits of free cyclic submodules generated by outliers in a specific matrix ring, highlighting different types of rings based on outlier properties.

Findings

Classified all orbits of free cyclic submodules in the matrix ring

Identified rings with outliers generating only torsion submodules

Provided examples illustrating different outlier behaviors

Abstract

We discuss the free cyclic submodules over an associative ring $R$ with unity. Special attention is paid to those, which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular $3$x$3$ matrices over a field $F$ under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.