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 Free cyclic codes over finite local rings Bull. Korean Math. Soc. 2006 Vol. 43, No. 4, 723-735 Published online December 1, 2006 Sung Sik Woo Ewha Women's University Abstract : In [2] it was shown that a 1-generator quasi-cyclic code $C$ of length $n=ml$ of index $l$ over $\Bbb Z_4$ 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 $\Bbb Z_{p^k}$ 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. Keywords : free modules over a finite commutative rings, separable extension of local rings, cyclic codes over $\Bbb Z_{p^k}$ MSC numbers : 13C10 Downloads: Full-text PDF