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.

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.