The purpose of this paper is to study the structure of quadratic residue codes over the ring $R=\mathbb{F}_{p^r}+u_1\mathbb{F}_{p^r}+u_2 \mathbb{F}_{p^r}+...+u_t \mathbb{F}_{p^r}$, where $r, t \geq 1$ and $p$ is a prime number. First we survey known results on quadratic residue codes over $\mathbb{F}_{p^r}$ and give general properties with quadratic residue codes over $R$. We introduce Gray map from $R$ to $\mathbb{F}^n_{p^r}$ and study a more details about the quadratic residue codes over the ring for $p=2, 3$ and investigate their Gray images over this ring. Finally, we obtain a number of Hermitian self-dual codes over $R$ in the case where $p=2$ and $p=3$ if $r$ is an even number or $r=1$ and $r$ is an even number, respectively, where $t$ is an odd number.
