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The purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Sole, and Qian for cyclic Z"4-codes to cyclic Z"p"^"m-codes. Second, we establish connections between this new development and the results on cyclic Z"p"^"m-codes obtained by Calderbank and Sloane. We produce generators for the cyclic Z"p"^"m-codes which are analogs to those for cyclic Z"4-codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ringR"n= Z"p"^"m[x]/(x^n- 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic Z"p"^"m-codes are produced from the generators of the corresponding cyclic Z"p"^"m-codes. In addition, we also obtain generators for the cyclicp^m-ary codes induced from the idempotent generators for cyclicp-ary codes.