On cyclic codes over $\mathbb{Z}_q+u\mathbb{Z}_q$

Let $R=\mathbb{Z}q+u\mathbb{Z}q$, where $q=p^s$ and $u^2=0$. In this paper, some structural properties of cyclic codes over the ring $R$ are considered. A necessary and sufficient condition for cyclic codes over the ring $R$ to be free is obtained and a BCH-type bound on the minimum Hamming distance for them is also given.