Abstract : In [6, 8], we showed that any ideal of $\mathbb{Z}_4[X]/(X^{l}-1)$ is generated by at most two polynomials of the `standard' forms when $l$ is even. The purpose of this paper is to find the `standard' generators of the cyclic codes over $\mathbb{Z}_{p^a}$ of length a multiple of $p$, namely the ideals of $\mathbb{Z}_{p^a}[X]/(X^l-1)$ with an integer $l$ which is a multiple of $p$. We also find an explicit description of their duals in terms of the generators when $a=2$.

Keywords : cyclic code over $\mathbb{Z}_{p^a}$, ideals of $ \mathbb{Z}_{p^n}[X]/(X^l-1)$