Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions

TL;DR

This paper classifies vectors and cyclic submodules over the ring of ternions, linking free cyclic submodules generated by non-unimodular vectors to lines in projective spaces, with explicit formulas in finite cases.

Contribution

It provides a complete classification of vectors and cyclic submodules over ternion rings and connects non-unimodular free cyclic submodules with projective space lines, including explicit formulas for finite fields.

Findings

Classification of vectors into 5 + |F| orbits under GL_{n+1}(R)

Explicit formulas for counting non-unimodular free cyclic submodules in finite fields

Connection between non-unimodular submodules and lines in projective spaces

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 $R^{n+1}$, $n \geq 1$, 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 $\PG(n,q)$, $n\geq 2$, in terms of vectors and non-unimodular free cyclic submodules of $R^{n+1}$.