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Left Co-K\"othe Rings and Their Characterizations

Shadi AsgariMahmood BehboodiSomayeh Khedrizadeh
Dec 2022
摘要
The classical K\"othe's problem posed by G. K\"othe in 1935 asks to describethe rings $R$ such that every left $R$-module is a direct sum of cyclic modules(are known as left K\"othe rings). K\"othe, Cohen and Kaplansky solved thisproblem for all commutative rings (that are Artinian principal ideal rings).During the years 1962 to 1965, Kawada solved the K\"othe's problem for basicfnite-dimensional algebras. But, so far, the K\"othe's problem was open in thenon-commutative setting. Recently, in the paper ["Several Characterizations ofLeft K\"othe Rings", submitted], we brook the class of left K\"othe rings intothree categories of nested: left K\"othe rings, strongly left K\"othe rings andvery strongly left K\"othe rings, and then, we solved the K\"othe's problem bygiving several characterizations of these rings in terms of describing theindecomposable modules. In this paper, we will introduce the Morita duality ofthese notions as co-K\"othe rings, left co-K\"othe rings and strongly leftco-K\"othe rings, and then, we give several structural characterizations foreach of them.
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