A ring $\unicode[STIX]{x1D6EC}$ is called right Köthe if every right $\unicode[STIX]{x1D6EC}$ -module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right Köthe rings in terms of their Coxeter valued quivers. We also give a characterization of basic right Köthe rings with radical square zero. Therefore, we give a solution to Köthe’s problem in these two cases.

中文翻译：

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Coxeter 图和 Köthe 问题

如果每个右$\unicode[STIX]{x1D6EC}$ -module 都是循环模块的直接和，则环$\unicode[STIX]{x1D6EC}$被称为右 Köthe 。在本文中，我们根据 Coxeter 值的箭袋给出了基本遗传权利 Köthe 环的特征。我们还给出了具有根方零的基本正确 Köthe 环的表征。因此，我们给出了这两种情况下 Köthe 问题的解决方案。