Linear and cyclic codes over some special rings

In this paper, we describe linear and cyclic codes over the rings of the form $R_{s,p}={\mathbb{Z}}_p[u]\mathrm{/}\left(f\left(u\right)\mathrm{/}(u-s)\right)$, where $p$ is a prime number and $f\left(u\right)=u^p-u$, with $s\in \{0,1,\mathrm{.}\mathrm{.}\mathrm{.},p-1\}$.