Several Characterizations of Left Köthe Rings

We study the classical Köthe’s problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring $R$ is called a $leftK\ddot{o}thering$ if every left $R$-module is a direct sum of cyclic $R$-modules. In 1951, Cohen and Kaplansky proved that all commutative K{ö}the rings are Artinian principal ideal rings. During the years 1962 to 1965, Kawada solved the Köthe’s problem for basic fnite-dimensional algebras: Kawada’s theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has square-free socle and square-free top, and describes the possible indecomposable modules. But, so far, the Köthe’s problem is open in the non-commutative setting. In this paper, we break the class of left K{ö}the rings into three categories of nested: $leftK\ddot{o}therings$, $stronglyleftK\ddot{o}therings$ and $verystronglyleftK\ddot{o}therings$, and then, we solve the Köthe’s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of Köthe-Cohen-Kaplansky theorem.