Coxeter diagrams and the K\"othe's problem

A ring $\mathrm{\Lambda }$ is called right K"othe if every right $\mathrm{\Lambda }$-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.