Connections between representation-finite and Köthe 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 RkR. 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.