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| Paper IPM / M / 17340 |
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| Abstract: | |||||
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We study classical K�?�öthe�â??s problem, concerning the structure of non-commutative rings with the property that: �â??every leftmodule is a direct sum of cyclicmodules".In 1934, K�?�öthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring R is called a left K�?�öthe ring if every left R-module is a direct sum of cyclic R-modules. In 1951, Cohen and Kaplansky proved that all commutative K�?�öthe rings are Artinian principal ideal rings. During the years 1961�â??1965, Kawada solved K�?�öthe�â??s problem for basic finitedimensional algebras: Kawada�â??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, K�?�öthe�â??s problem is open in the non-commutative setting. In this paper, 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 solve K�?�öthe�â??s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of K�?�öthe�â??Cohen�â??Kaplansky theorem.
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