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We study classical KÃ??Ã?ÃÂ¶theÃ?ÃÂ¢??s problem, concerning the structure of non-commutative rings with the property that: Ã?ÃÂ¢??every leftmodule is a direct sum of cyclicmodules".In 1934, KÃ??Ã?ÃÂ¶the showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring R is called a left KÃ??Ã?ÃÂ¶the ring if every left R-module is a direct sum of cyclic R-modules. In 1951, Cohen and Kaplansky proved that all commutative KÃ??Ã?ÃÂ¶the rings are Artinian principal ideal rings. During the years 1961Ã?ÃÂ¢??1965, Kawada solved KÃ??Ã?ÃÂ¶theÃ?ÃÂ¢??s problem for basic finitedimensional algebras: KawadaÃ?ÃÂ¢??s theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has a square-free socle and a square-free top, and describes the possible indecomposable modules. But, so far, KÃ??Ã?ÃÂ¶theÃ?ÃÂ¢??s problem is open in the non-commutative setting. In this paper, we classified left KÃ??Ã?ÃÂ¶the rings into three classes one contained in the other: left KÃ??Ã?ÃÂ¶the rings, strongly left KÃ??Ã?ÃÂ¶the rings and very strongly left KÃ??Ã?ÃÂ¶the rings, and then, we solve KÃ??Ã?ÃÂ¶theÃ?ÃÂ¢??s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of KÃ??Ã?ÃÂ¶theÃ?ÃÂ¢??CohenÃ?ÃÂ¢??Kaplansky theorem.
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