# Coxeter diagrams and the K\"othe's problem

## TL;DR

This paper characterizes certain classes of right K"othe rings using Coxeter valued quivers and provides solutions to K"othe's problem for rings with radical square zero.

## Contribution

It offers a new characterization of basic hereditary right K"othe rings and those with radical square zero via Coxeter diagrams, solving K"othe's problem in these cases.

## Key findings

- Characterization of hereditary right K"othe rings using Coxeter valued quivers
- Characterization of right K"othe rings with radical square zero
- Solution to K"othe's problem for these specific cases

## Abstract

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.

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Source: https://tomesphere.com/paper/1812.06642