  Motivated by physical-layer network coding, this paper considers
communication in multiplicative matrix channels over finite chain rings. Such
channels are defined by the law $Y =A X$, where $X$ and $Y$ are the input and
output matrices, respectively, and $A$ is called the transfer matrix. It is
assumed a coherent scenario in which the instances of the transfer matrix are
unknown to the transmitter, but available to the receiver. It is also assumed
that $A$ and $X$ are independent. Besides that, no restrictions on the
statistics of $A$ are imposed. As contributions, a closed-form expression for
the channel capacity is obtained, and a coding scheme for the channel is
proposed. It is then shown that the scheme can achieve the capacity with
polynomial time complexity and can provide correcting guarantees under a
worst-case channel model. The results in the paper extend the corresponding
ones for finite fields.
