Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions

Abstract

Given a ring of ternions $R$, i. e., a ring isomorphic to that of upper triangular $2\times 2$ matrices with entries from an arbitrary commutative field $F$, a complete classification is performed of the vectors from the free left $R$-module $R^{n+1}$, $n\ge 1$, 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 $\GL_{n+1}(R)$. Particular attention is paid to {\it free} cyclic submodules generated by \emph{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 $\PG(n,q)$, $n\ge 2$, in terms of vectors and non-unimodular free cyclic submodules of $R^{n+1}$.

10 pages, 1 figure

References in corpus (3)

• Projective Ring Line of an Arbitrary Single Qudit
• A Classification of the Projective Lines over Small Rings
• Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions

Cited by in corpus (8)

• Geometry of free cyclic submodules over ternions
• Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?
• A point model for the free cyclic submodules over ternions
• A Jacobson Radical Decomposition of the Fano-Snowflake Configuration
• Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices
• Free Cyclic Submodules and Non-Unimodular Vectors
• Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules
• Projective Ring Lines and Their Generalisations