In this paper,\,we study linear and cyclic codes over the ring $F_2+uF_2+vF_2$.\,The ring $F_2+uF_2+vF_2$ is the smallest non-Frobenius ring.\,We characterize the structure of cyclic codes over the ring $R=F_2+uF_2+vF_2$ using of the work {\co Abualrub and Saip}\,(Des.\,Codes Crypt.\,42\,:273-287,2007).\,We study the rank and dual of cyclic codes of odd length over this ring.\,Specially,\,we show that the equation $|C||C^\bot|= |R|^n$ does not hold in general for a cyclic code $C$ of length $n$ over this ring.\,We also obtain some optimal binary codes as the images of cyclic codes over the ring $F_2+uF_2+vF_2$ under a Gray map,\,which maps Lee weights to Hamming weights.\,Finally,\,we give a condition for cyclic codes over $R$ that contains its dual and find quantum codes over $F_2$ from cyclic codes over the ring $F_2+uF_2+vF_2$.
