  Let $\texttt{R}$ be a commutative finite chain ring of invariants $(q,s).$ In
this paper, the trace representation of any free cyclic $\texttt{R}$-linear
code of length $\ell,$ is presented, via the $q$-cyclotomic cosets modulo
$\ell,$ when $\texttt{gcd}(\ell, q) = 1.$ The lattice
$\left(\texttt{Cy}(\texttt{R},\ell), +, \cap\right)$ of cyclic
$\texttt{R}$-linear codes of length $\ell,$ is investigated. A lower bound on
the Hamming distance of cyclic $\texttt{R}$-linear codes of length $\ell,$ is
established. When $q$ is even, a family of MDS and self-orthogonal
$\texttt{R}$-linear cyclic codes, is constructed.
