Geometries on σ-Hermitian matrices

Ring geometry and the geometry of matrices meet naturally at the ring R := Kn×n of n× n matrices with entries in a (not necessarily commutative) field K. Our aim is to strengthen the interaction between these disciplines. Below we sketch some results from either side, even though not in their most general form, but in a way which is tailored for our needs. Let us start with ring geometry, where we follow [7] and [10]: Consider the free left R-module R2 and the group GL2(R) = GL2n(K) of invertible 2 × 2-matrices with entries in R. A pair (A, B) ∈ R2 is called admissible if there exists a matrix in GL2(R) with (A, B) being its first row. The projective line over R, in symbols P(R), is the set of cyclic submodules R(A, B) for all admissible pairs (A, B) ∈ R2. Two admissible pairs represent the same point precisely when they are left-proportional by a unit in R, i. e., a matrix from GLn(K). Conversely, if R(A′, B′) = R(A, B) for some pair (A′, B′) ∈ R2 and an admissible pair (A, B) ∈ R2 then (A′, B′) is admissible too [3, Proposition 2.2]. By [2], the projective line over R allows the following description which is not available for arbitrary rings, as it makes use of the left row rank of a matrix X over K (in symbols: rank X):