One Cyclic Codes over $\mathbb{F}_{p^k} + v\mathbb{F}_{p^k} + v^2\mathbb{F}_{p^k} + ... + v^r\mathbb{F}_{p^k}$

In this paper, we investigate cyclic code over the ring $\mathbb{F}{p^k} + v\mathbb{F}{p^k} + v^{2\mathbb{F}_{pk}} + ... + v^{r\mathbb{F}_{pk}$,} where $v^{r+1}=v$, $p$ a prime number, $r>1$ and $\gcd(r,p)=1$, we prove as generalisation of P. Sol\'e et al. in 2015 that these codes are principally generated, give generator polynomial and idempotent depending on idempotents over this ring as response to an open problem related by J. QIAN et al. in 2005. we also give a gray map and proprieties of the related dual code.