Coxeter diagrams and the K\'othe's problem

A ring $\Lambda$ is called right K\"othe if every right $\Lambda$-module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right K\"othe rings in terms of their Coxeter valued quivers. Also we give a characterization of basic right K\"othe rings with radical square zero. Therefore we give a solution of the K\"othe's problem in this cases.
Comment: to appear in Canad. J. Math