  In this paper, we solve the reversibility problem for DNA codes over the
non-chain ring $R_{k,s}=\mathbb{F}_{4^{2k}}[u_1,...,u_{s}]/< u_1^2-u_1,...,
u_s^2-u_s>$. We define an automorphism $\theta$ over $R_{k,s}$ which help us
both find the idempotent decomposition of $R_{k,s}$ and solve the reversibility
problem via skew cyclic codes. Moreover, we introduce a generalized Gray map
that preserves DNA reversibility.
