On Projective Geometry Over Simple Matrix Rings

Abstract

This paper investigating the relationship between projective geometry over a ring R and those over a simple matrix ring , demonstrating these geometries are essentially equivalent. The study extends the fundamental theorem of projective geometry modules, where is a division ring, traditionally applicable to vector spaces over a division rings, to the border context of modules over rings. Specifically, it focuses on the lattice structure formed by all R-sub-modules of an R-module., which defines the projective geometry in this setting. The equivalence between projective geometries over R and is established, highlighting the structural similarities and providing a framework for understanding projective geometry in the more general context of modules. This work contributes to the border understanding of   projective geometry, particularly in non-classical settings, and aligns with the 2020 AMS subject Classification 16S10.