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| Paper IPM / M / 17342 |
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| Abstract: | |||||
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K�ötheâ??s classical problem posed by G. K�öthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left K�öthe rings). K�öthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved K�ötheâ??s problem for basic finite-dimensional algebras. But, so far, K�ötheâ??s problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left K�öthe rings, Rev.
Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)], we classified left K�öthe rings into three classes one contained in the other: left K�öthe rings, strongly left K�öthe rings and very strongly left K�öthe rings, and then, we solved K�ötheâ??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 co-K�öthe rings, strongly left co-K�öthe
rings and very strongly left co-K�öthe rings, and then, we give several structural characterizations for each of them.
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