This website requires JavaScript.

Several Characterizations of Left K\"othe Rings

Shadi AsgariMahmood BehboodiSomayeh Khedrizadeh
Jun 2022
摘要
We study the classical K\"othe's problem, concerning the structure ofnon-commutative rings with the property that: ``every left module is a directsum of cyclic modules". In 1934, K\"othe showed that left modules over Artinianprincipal ideal rings are direct sums of cyclic modules. A ring $R$ is called a${\it left~K\"othe~ring}$ if every left $R$-module is a direct sum of cyclic$R$-modules. In 1951, Cohen and Kaplansky proved that all commutative K{\"o}therings are Artinian principal ideal rings. During the years 1962 to 1965, Kawadasolved the K\"othe's problem for basic fnite-dimensional algebras: Kawada'stheorem characterizes completely those finite-dimensional algebras for whichany indecomposable module has square-free socle and square-free top, anddescribes the possible indecomposable modules. But, so far, the K\"othe'sproblem is open in the non-commutative setting. In this paper, we break theclass of left K{\"o}the rings into three categories of nested: ${\itleft~K\"othe~rings}$, ${\it strongly~left~K{\"o}the~rings}$ and ${\itvery~strongly~left~K{\"o}the~rings}$, and then, we solve the K\"othe's problemby giving several characterizations of these rings in terms of describing theindecomposable modules. Finally, we give a new generalization ofK\"othe-Cohen-Kaplansky theorem.
展开全部
图表提取

暂无人提供速读十问回答

论文十问由沈向洋博士提出,鼓励大家带着这十个问题去阅读论文,用有用的信息构建认知模型。写出自己的十问回答,还有机会在当前页面展示哦。

Q1论文试图解决什么问题?
Q2这是否是一个新的问题?
Q3这篇文章要验证一个什么科学假设?
0
被引用
笔记
问答