In this paper, cyclic codes of arbitrary length n over the ring R2,m are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F2m -basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one correspondence between cyclic codes of length 2n , n odd, over the ring R k−1,m and cyclic codes of length n over the ring R k,m . By determining the complete structure of cyclic codes of length 2 over R2,m , a mass formula for the number of these codes is given. Using this and the mentioned correspondence, the number of ideals of the rings R2,m and R3,m is determined. As a corollary, the number of cyclic codes of odd length n over the rings R2,m and R3,m is obtained.

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