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A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules which are homomorphic image of R k R. In this paper, we give a characterization of left k-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe–Cohen–Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k-cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Köthe ring if and only if R is an artinian left multiplicity-free top ring.