50 years of Finite Geometry, the "geometries over finite rings" part

TL;DR

This paper reviews the development of finite geometries over finite rings beyond fields, consolidating existing results to encourage further research in this less-explored area of finite geometry.

Contribution

It compiles and synthesizes scattered results on finite rings and their associated geometries, providing a comprehensive overview without introducing new findings.

Findings

Finite rings give rise to interesting geometries.

Collected results can stimulate future research.

Focus on geometries over non-division finite rings.

Abstract

Whereas for a substantial part, Finite Geometry during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, several important classes of finite rings give rise to interesting geometries. In this paper we bring together some results, scattered over the literature, concerning finite rings and plane projective geometry over such rings. It doesn't contain new material, but by collecting stuff in one place, we hope to stimulate further research in this area for at least another 50 years of Finite Geometry.