Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions

Given a ring of ternions R, i. e., a ring isomorphic to that of upper triangular 2×2 matrices with entries from an arbitrary commutative field F , a complete classification is performed of the vectors from the free left R-module R, n ≥ 1, and of the cyclic submodules generated by these vectors. The vectors fall into 5 + |F | and the submodules into 6 distinct orbits under the action of the general linear group GLn+1(R). Particular attention is paid to free cyclic submodules generated by non-unimodular vectors, as these are linked with the lines of PG(n, F ), the n-dimensional projective space over F . In the finite case, F = GF(q), explicit formulas are derived for both the total number of non-unimodular free cyclic submodules and the number of such submodules passing through a given vector. These formulas yield a combinatorial approach to the lines and points of PG(n, q), n ≥ 2, in terms of vectors and non-unimodular free cyclic submodules of R.