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Given a ring of ternions $R$, i. e., a ring isomorphic to that of uppertriangular $2\times 2$ matrices with entries from an arbitrary commutativefield $F$, a complete classification is performed of the vectors from the freeleft $R$-module $R^{n+1}$, $n \geq 1$, and of the cyclic submodules generatedby these vectors. The vectors fall into $5 + |F|$ and the submodules into 6distinct 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 ofnon-unimodular free cyclic submodules and the number of such submodules passingthrough a given vector. These formulas yield a combinatorial approach to thelines and points of $\PG(n,q)$, $n\geq 2$, in terms of vectors andnon-unimodular free cyclic submodules of $R^{n+1}$.

Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions