Geometry of free cyclic submodules over ternions

Abstract

Abstract. Given the algebra T of ternions (upper triangular 2 × 2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T 2 . This set of points can be represented as a set of planes in the projective space over F 6 . We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F -linear antiautomorphism, the plane model of our projective line does not admit any duality.

On the history of ring geometry (with a thematical overview of literature)

“…Some papers by Bilo and Depunt [B01], Hubaut [B08, B09] and Thas [B10, B11] also deal with projective lines over rings (see section 4). They were followed by Havlicek et al in [K26,K27,K28].…”

Free Cyclic Submodules in the Context of the Projective Line

“…Hence aR+bR = αxR + αyR = α(xR + yR) = αR, which completes the proof. Example 3 [14,15]. Consider the ring T of ternions over the commutative field F , i.e.…”

Projective ring lines and their generalisations

“…A wealth of further references can be found in [2], [11], [19], [24], [28], [35], [37], and [38]. Refer to [12], [13], [17], [20], [21], [22], and [32] for deviating definitions of projective lines which we cannot present here.…”