A ring representing a finite projective geometry

We define a ring that represents a finite projective geometry of any dimension, and study various algebraic properties of the ring and categorical behavior between the ring and the geometry. In addition to basic ring theoretic properties, we give a sufficient condition for a ring to represent a projective geometry, and study relations between the ring homomorphisms and the geometry homomorphisms. Then the decomposition and the inertia groups of the automorphisms of the ring are discussed. We find an interesting relation between the Hilbert function of the ring and the ovals of the projective plane in certain cases. We also study a reasonable Z-coalgebra structure on the ring representing a projective plane.