# Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices

## TL;DR

This paper explores the connection between affine and projective planes and the projective lines over rings of lower triangular matrices, revealing their structure and extension properties for specific matrix sizes.

## Contribution

It demonstrates how projective lines over certain matrix rings generate affine planes and how these can be extended to projective planes, providing a new geometric perspective on these algebraic structures.

## Key findings

- Affine planes of order q are generated from projective lines over T_n(q).
- Each affine plane extends uniquely to a projective plane of order q.
- The structure applies to matrices of sizes n=2,3, with potential generalization.

## Abstract

Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane can be extended to the projective plane of order $q$, with the `line at infinity' being represented by those free cyclic submodules of ${^2T_n(q)}$ that are generated by non-unimodular pairs. Our approach can straightforwardly be adjusted to address the case of arbitrary $n$.