  Let $\mathbb{F}_q$ be a finite field of order $q$, a prime power integer such
that $q=et+1$ where $t\geq 1,e\geq 2$ are integers. In this paper, we study
cyclic codes of length $n$ over a non-chain ring
$R_{e,q}=\mathbb{F}_q[u]/\langle u^e-1\rangle$. We define a Gray map $\varphi$
and obtain many { maximum-distance-separable} (MDS) and optimal
$\mathbb{F}_q$-linear codes from the Gray images of cyclic codes. Under certain
conditions we determine { linear complementary dual} (LCD) codes of length $n$
when $\gcd(n,q)\neq 1$ and $\gcd(n,q)= 1$, respectively. It is proved that { a}
cyclic code $\mathcal{C}$ of length $n$ is an LCD code if and only if its Gray
image $\varphi(\mathcal{C})$ is an LCD code of length $4n$ over $\mathbb{F}_q$.
Among others, we present the conditions for existence of free and non-free LCD
codes. Moreover, we obtain many optimal LCD codes as the Gray images of
non-free LCD codes over $R_{e,q}$.
