  Due to their rich algebraic structure, cyclic codes have a great deal of
significance amongst linear codes. Duadic codes are the generalization of the
quadratic residue codes, a special case of cyclic codes. The $m$-adic residue
codes are the generalization of the duadic codes. The aim of this paper is to
study the structure of the $m$-adic residue codes over the quotient ring
$\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v}
\right\rangle }}$. We determine the idempotent generators of the $m$-adic
residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle
{{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic
residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle
{{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings.
Furthermore, we derive a condition for $m$-adic residue codes over
$\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v}
\right\rangle }}$ to contain their dual. By making use of a
preserving-orthogonality Gray map, we construct a family of quantum error
correcting codes from the Gray images of dual-containing $m$-adic residue codes
over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v}
\right\rangle }}$ and give some examples to illustrate our findings.
