The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of the residue field R~. Some cases where n is divisible by the characteristic of the residue field R~ are also considered. Namely, the structure of negacyclic codes of length 2/sup t/ over /spl Zopf//sub 2//sup m/ and that of their duals are derived.