In this paper, we study cyclic codes of length n over R = Z q + uZ q , u 2 = 0, where q is a power of a prime p and (n, p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of x n − 1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z 2r p is defined. Using Magma, some good cyclic codes of length 4 over Z 9 + uZ 9 are obtained.