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In this paper, we investigate cyclic code over the ring $\mathbb{F}_{p^k} +v\mathbb{F}_{p^k} + v^2\mathbb{F}_{p^k} + ... + v^r\mathbb{F}_{p^k}$, where$v^{r+1}=v$, $p$ a prime number, $r>1$ and $\gcd(r,p)=1$, we prove asgeneralisation of P. Sol\'e et al. in 2015 that these codes are principallygenerated, give generator polynomial and idempotent depending on idempotentsover this ring as response to an open problem related by J. QIAN et al. in2005. we also give a gray map and proprieties of the related dual code.

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