  Let $p$ be a prime integer, $n,s\geq 2$ be integers satisfying ${\rm
gcd}(p,n)=1$, and denote $R=\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$. Then
$R$ is a local non-principal ideal ring of $p^{2s}$ elements. First, the
structure of any cyclic code over $R$ of length $n$ and a complete
classification of all these codes are presented. Then the cardinality of each
code and dual codes of these codes are given. Moreover, self-dual cyclic codes
over $R$ of length $n$ are investigated. Finally, we list some optimal
$2$-quasi-cyclic self-dual linear codes over $\mathbb{Z}_4$ of length $30$ and
extremal $4$-quasi-cyclic self-dual binary linear $[60,30,12]$ codes derived
from cyclic codes over $\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle$ of length
$15$.
