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Paper IPM / M / 17342  


Abstract:  
KÃ?ÃÂ¶theÃÂ¢??s classical problem posed by G. KÃ?ÃÂ¶the in 1935 asks to describe the rings R such that every left Rmodule is a direct sum of cyclic modules (these rings are known as left KÃ?ÃÂ¶the rings). KÃ?ÃÂ¶the, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved KÃ?ÃÂ¶theÃÂ¢??s problem for basic finitedimensional algebras. But, so far, KÃ?ÃÂ¶theÃÂ¢??s problem was open in the noncommutative setting. Recently, in the paper [Several characterizations of left KÃ?ÃÂ¶the rings, Rev.
Real Acad. Cienc. Exactas Fis. Nat. Ser. AMat. (2023) (to appear)], we classified left KÃ?ÃÂ¶the rings into three classes one contained in the other: left KÃ?ÃÂ¶the rings, strongly left KÃ?ÃÂ¶the rings and very strongly left KÃ?ÃÂ¶the rings, and then, we solved KÃ?ÃÂ¶theÃÂ¢??s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left coKÃ?ÃÂ¶the rings, strongly left coKÃ?ÃÂ¶the
rings and very strongly left coKÃ?ÃÂ¶the rings, and then, we give several structural characterizations for each of them.
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