  Additive cyclic codes over Galois rings were investigated in previous works.
In this paper, we investigate the same problem but over a more general ring
family, finite commutative chain rings. When we focus on non-Galois finite
commutative chain rings, we observe two different kinds of additivity. One of
them is a natural generalization of the previous studies, whereas the other one
has some unusual properties especially while constructing dual codes. We
interpret the reasons of such properties and illustrate our results giving
concrete examples.
