  Let $\mathbb{F}_q$ be the finite field of $q=p^m$ elements where $p$ is a
prime and $m$ is a positive integer. This paper considers
$(\gamma,\Delta)$-cyclic codes over a class of finite commutative non-chain
rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle
v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $\gamma$ is an automorphism of
$\mathscr{R}_{q,s}$, $\Delta$ is a $\gamma$-derivation of $\mathscr{R}_{q,s}$
and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a
$(\gamma,\Delta)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the
direct sum of $(\theta,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$,
where $\theta$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a
$\theta$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient
conditions for both $(\gamma,\Delta)$-cyclic and $(\theta,\Im)$-cyclic codes to
contain their Euclidean duals are established. Finally, we obtain many quantum
codes by applying the dual containing criterion on the Gray images of these
codes. The obtained codes have better parameters than those available in the
literature.
