In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over $mathbb{Z}_4$ is free if C is generated by a polynomial which divides $X^m-1$. In this paper, we prove that a necessary and sufficient condition for a cyclic code over $mathbb{Z}_pk$ of length m to be free is that it is generated by a polynomial which divides $X^m-1$. We also show that this can be extended to finite local rings with a principal maximal ideal.