  The structure of multivariate semisimple codes over a finite chain ring $R$
is established using the structure of the residue field $\bar R$. Multivariate
codes extend in a natural way the univariate cyclic and negacyclic codes and
include some non-trivial codes over $R$. The structure of the dual codes in the
semisimple abelian case is also derived and some conditions on the existence of
selfdual codes over $R$ are studied.
