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Paper   IPM / M / 17342
School of Mathematics
  Title:   Left co-Kothe rings and their characterizations
1.  Shadi Asgari
2.  Mahmood Behboodi (Joint with S. Khedrizadeh)
  Status:   Published
  Journal: Comm. Algebra
  Vol.:  51
  Year:  2023
  Pages:   1-20
  Supported by:  IPM
K�¶theâ??s classical problem posed by G. K�¶the 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�¶the rings). K�¶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 K�¶theâ??s problem for basic finite-dimensional algebras. But, so far, K�¶theâ??s problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left K�¶the rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)], we classified left K�¶the rings into three classes one contained in the other: left K�¶the rings, strongly left K�¶the rings and very strongly left K�¶the rings, and then, we solved K�¶theâ??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�¶the rings, strongly left co-K�¶the rings and very strongly left co-K�¶the rings, and then, we give several structural characterizations for each of them.

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