  For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be the
finite field of characteristic $p$, and
$\mathfrak{R}_l:=\mathbb{F}_{p^m}[v]/\langle v^l-v\rangle$ be a non-chain ring.
In this paper, we study the $(\sigma,\delta)$-cyclic codes over
$\mathfrak{R}_l$. Further, we study the application of these codes in finding
DNA codes. Towards this, we first define a Gray map to find classical codes
over $\mathbb{F}_{p^m}$ using codes over the ring $\mathfrak{R}_l$. Later, we
find the conditions for a code to be reversible and a DNA code using $(\sigma,
\delta)$-cyclic code. Finally, this algebraic method provides many classical
and DNA codes of better parameters.
