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• W4383755621 abstract "We study 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 left Köthe ring if every left R-module is a direct sum of cyclic R-modules. In 1951, Cohen and Kaplansky proved that all commutative Köthe rings are Artinian principal ideal rings. During the years 1961–1965, Kawada solved Köthe’s problem for basic finite-dimensional algebras: Kawada’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, Köthe’s problem is open in the non-commutative setting. In this paper, we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solve 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." @default.
• W4383755621 created "2023-07-11" @default.
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• W4383755621 date "2023-07-10" @default.
• W4383755621 modified "2023-10-01" @default.
• W4383755621 title "Several characterizations of left Köthe rings" @default.
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• W4383755621 doi "https://doi.org/10.1007/s13398-023-01477-9" @default.
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