Repeated Root Constacyclic Codes of Length $m{p}^s$ over ${\mathbb{F}}_{p^r}+u{\mathbb{F}}_{p^r}+\mathrm{.}\mathrm{.}\mathrm{.}+u^{e-1}{\mathbb{F}}_{p^r}$

We give the structure of $\lambda$-constacyclic codes of length $p^{s}m$ over $R={\mathbb{F}}_{p^r}+u{\mathbb{F}}_{p^r}+\mathrm{.}\mathrm{.}\mathrm{.}+u^{e-1}{\mathbb{F}}_{p^r}$ with $\lambda \in {\text{F}}_{p^r}^{\ast }$. We also give the structure of $\lambda$-constacyclic codes of length $p^{s}m$ with $\lambda=\alpha_1+u\alpha_2+…+u^{e-1} \alpha_{e-1}$,where$\alpha_1,\alpha_2 \neq 0$ and study the self-duality of these codes.