  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 $_{R}R^k$. In this
paper, we give a characterization of left $k$-cyclic rings. As a consequence,
we give a characterization of left K\"othe rings, which is a generalization of
K\"othe-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 K\"othe ring if and only if $R$ is an
artinian left multiplicity-free top ring.
