Connections between Representation-Finite and K\"othe Rings

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.