  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 $R^{n+1}$.
