The results of J. F. Qiann et al. [4] on $(1-\gamma)$-cyclic codes over
finite chain rings of nilpotency index 2 are extended to $(1-\gamma^e)$-cyclic
codes over finite chain rings of arbitrary nilpotency index $e+1$. The Gray map
is introduced for this type of rings. We prove that the Gray image of a linear
$(1 - \gamma^{e})$-cyclic code over a finite chain ring is a distance-invariant
quasi-cyclic code over its residue field. When the length of codes and the
characteristic of a ring are relatively prime, the Gray images of a linear
cyclic code and a linear $(1+\gamma^e)$-cyclic code are permutatively to
quasi-cyclic codes over its residue field.