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# Several Characterizations of Left K\"othe Rings

Jun 2022

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.

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