  Though network coding is traditionally performed over finite fields, recent
work on nested-lattice-based network coding suggests that, by allowing network
coding over certain finite rings, more efficient physical-layer network coding
schemes can be constructed. This paper considers the problem of communication
over a finite-ring matrix channel $Y = AX + BE$, where $X$ is the channel
input, $Y$ is the channel output, $E$ is random error, and $A$ and $B$ are
random transfer matrices. Tight capacity results are obtained and simple
polynomial-complexity capacity-achieving coding schemes are provided under the
assumption that $A$ is uniform over all full-rank matrices and $BE$ is uniform
over all rank-$t$ matrices, extending the work of Silva, Kschischang and
K\"{o}tter (2010), who handled the case of finite fields. This extension is
based on several new results, which may be of independent interest, that
generalize concepts and methods from matrices over finite fields to matrices
over finite chain rings.
